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LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners Chapter LINEAR TRANSFORMATIONS 8.1 Euclidean Linear Transformations By a transformation from Rn into Rm , we mean a function of the type T : Rn → Rm , with domain Rn and codomain Rm For every vector x ∈ Rn , the vector T (x) ∈ Rm is called the image of x under the transformation T , and the set R(T ) = {T (x) : x ∈ Rn }, of all images under T , is called the range of the transformation T Remark For our convenience later, we have chosen to use R(T ) instead of the usual T (Rn ) to denote the range of the transformation T For every x = (x1 , , xn ) ∈ Rn , we can write T (x) = T (x1 , , xn ) = (y1 , , ym ) Here, for every i = 1, , m, we have yi = Ti (x1 , , xn ), (1) where Ti : Rn → R is a real valued function Definition A transformation T : Rn → Rm is called a linear transformation if there exists a real matrix   a11 a1n  A =  am1 amn Chapter : Linear Transformations page of 35 c Linear Algebra W W L Chen, 1997, 2008 such that for every x = (x1 , , xn ) ∈ Rn , we have T (x1 , , xn ) = (y1 , , ym ), where y1 = a11 x1 + + a1n xn , ym = am1 x1 + + amn xn , or, in matrix notation,    y1 a11    =   am1 ym   a1n x1    amn (2) xn The matrix A is called the standard matrix for the linear transformation T Remarks (1) In other words, a transformation T : Rn → Rm is linear if the equation (1) for every i = 1, , m is linear (2) If we write x ∈ Rn and y ∈ Rm as column matrices, then (2) can be written in the form y = Ax, and so the linear transformation T can be interpreted as multiplication of x ∈ Rn by the standard matrix A Definition A linear transformation T : Rn → Rm is said to be a linear operator if n = m In this case, we say that T is a linear operator on Rn Example 8.1.1 The linear transformation T : R5 → R3 , defined by the equations y1 = 2x1 + 3x2 + 5x3 + 7x4 − 9x5 , y2 = 3x2 + 4x3 y3 = x1 + 2x5 , + 3x3 − 2x4 , can be expressed in matrix form as    y1  y2  =  y3 3 −2   x1 −9  x2      x3    x4 x5  If (x1 , x2 , x3 , x4 , x5 ) = (1, 0, 1, 0, 1), then    y1  y2  =  y3 −2     −9   −2   1 =  ,   0  so that T (1, 0, 1, 0, 1) = (−2, 6, 4) Example 8.1.2 Suppose that A is the zero m × n matrix The linear transformation T : Rn → Rm , where T (x) = Ax for every x ∈ Rn , is the zero transformation from Rn into Rm Clearly T (x) = for every x ∈ Rn Example 8.1.3 Suppose that I is the identity n × n matrix The linear operator T : Rn → Rn , where T (x) = Ix for every x ∈ Rn , is the identity operator on Rn Clearly T (x) = x for every x ∈ Rn Chapter : Linear Transformations page of 35 c Linear Algebra W W L Chen, 1997, 2008 PROPOSITION 8A Suppose that T : Rn → Rm is a linear transformation, and that {e1 , , en } is the standard basis for Rn Then the standard matrix for T is given by A = ( T (e1 ) T (en ) ) , where T (ej ) is a column matrix for every j = 1, , n Proof This follows immediately from (2) 8.2 Linear Operators on R2 In this section, we consider the special case when n = m = 2, and study linear operators on R2 For every x ∈ R2 , we shall write x = (x1 , x2 ) Example 8.2.1 Consider reflection across the x2 -axis, so that T (x1 , x2 ) = (−x1 , x2 ) Clearly we have T (e1 ) = −1 and T (e2 ) = , and so it follows from Proposition 8A that the standard matrix is given by A= −1 0 It is not difficult to see that the standard matrices for reflection across the x1 -axis and across the line x1 = x2 are given respectively by A= 0 −1 and A= 1 Also, the standard matrix for reflection across the origin is given by A= −1 0 −1 We give a summary in the table below: Linear operator Equations Standard matrix Reflection across x2 -axis y1 = −x1 y2 = x2 −1 Reflection across x1 -axis y1 = x1 y2 = −x2 0 −1 Reflection across x1 = x2 y1 = x2 y2 = x1 1 Reflection across origin y1 = −x1 y2 = −x2 −1 0 −1 Example 8.2.2 For orthogonal projection onto the x1 -axis, we have T (x1 , x2 ) = (x1 , 0), with standard matrix A= Chapter : Linear Transformations 0 page of 35 c Linear Algebra W W L Chen, 1997, 2008 Similarly, the standard matrix for orthogonal projection onto the x2 -axis is given by 0 A= We give a summary in the table below: Linear operator Equations Standard matrix Orthogonal projection onto x1 -axis y1 = x1 y2 = 0 Orthogonal projection onto x2 -axis y1 = y2 = x2 0 Example 8.2.3 For anticlockwise rotation by an angle θ, we have T (x1 , x2 ) = (y1 , y2 ), where y1 + iy2 = (x1 + ix2 )(cos θ + i sin θ), and so y1 y2 = cos θ sin θ − sin θ cos θ x1 x2 It follows that the standard matrix is given by A= cos θ sin θ − sin θ cos θ We give a summary in the table below: Linear operator Equations Standard matrix y1 = x1 cos θ − x2 sin θ y2 = x1 sin θ + x2 cos θ Anticlockwise rotation by angle θ cos θ sin θ − sin θ cos θ Example 8.2.4 For contraction or dilation by a non-negative scalar k, we have T (x1 , x2 ) = (kx1 , kx2 ), with standard matrix A= k 0 k The operator is called a contraction if < k < and a dilation if k > 1, and can be extended to negative values of k by noting that for k < 0, we have k 0 k = −1 0 −1 −k 0 −k This describes contraction or dilation by non-negative scalar −k followed by reflection across the origin We give a summary in the table below: Linear operator Contraction or dilation by factor k Chapter : Linear Transformations Equations y1 = kx1 y2 = kx2 Standard matrix k 0 k page of 35 c Linear Linear Algebra Algebra W 2008 WW WL L Chen, Chen, 1997, 1997, 2006 2006 Example 8.2.5 For expansion or compression in the x11 -direction by a positive factor k, we have T (x11 , x22 ) = (kx11 , x22 ), with standard matrix A= k 0 This can be extended to negative values of k by noting that for k < 0, we have k k 00 00 11 = = −1 −1 00 00 11 −k −k 00 This This describes describes expansion expansion or or compression compression in in the the x x11 -direction -direction across the x -axis Similarly, for expansion or compression across the x2 -axis Similarly, for expansion or compression we we have have the the standard standard matrix matrix A A= = 11 00 00 k k 00 11 by by positive positive factor factor −k −k followed followed by by reflection reflection in the x -direction by a non-zero in the x2 -direction by a non-zero factor factor k, k, We We give give aa summary summary in in the the table table below: below: Linear Linear operator operator Equations Equations y111 = kx111 yy222 = =x x222 y111 = x111 yy222 = = kx kx222 Expansion Expansion or or compression compression in in x x111 -direction -direction Expansion Expansion or or compression compression in in x x222 -direction -direction Standard Standard matrix matrix k 00 11 00 k k Example Example 8.2.6 8.2.6 For For shears shears in in the the x x111 -direction -direction with with factor factor k, k, we we have have T T (x (x111 ,, x x222 )) = = (x (x111 + + kx kx222 ,, x x222 ), ), with with standard matrix standard matrix A A= = 11 k k 00 11 For For the the case case k k= = 1, 1, we we have have the the following following • • • • T T (k=1) (k=1) • • • • For For the the case case k = = −1, −1, we have have the the following following • • • • Chapter Chapter 8 :: Linear Linear Transformations Transformations T T (k=−1) (k=−1) • • • • page page 5 of of 35 35 c Linear Algebra W W L Chen, 1997, 2008 Similarly, for shears in the x2 -direction with factor k, we have standard matrix k A= We give a summary in the table below: Linear operator Equations Standard matrix Shear in x1 -direction y1 = x1 + kx2 y2 = x2 k Shear in x2 -direction y1 = x1 y2 = kx1 + x2 k Example 8.2.7 Consider a linear operator T : R2 → R2 which consists of a reflection across the x2 -axis, followed by a shear in the x1 -direction with factor and then reflection across the x1 -axis To find the standard matrix, consider the effect of T on a standard basis {e1 , e2 } of R2 Note that e1 = → −1 → −1 → −1 = T (e1 ), e2 = → → → −1 = T (e2 ), so it follows from Proposition 8A that the standard matrix for T is A= −1 −1 Let us summarize the above and consider a few special cases We have the following table of invertible linear operators with k = Clearly, if A is the standard matrix for an invertible linear operator T , then the inverse matrix A−1 is the standard matrix for the inverse linear operator T −1 Linear operator T Standard matrix A Inverse matrix A−1 Reflection across line x1 =x2 1 0 1 Expansion or compression in x1 −direction k 0 k −1 Expansion or compression in x2 −direction 0 k Shear in x1 −direction Shear in x2 −direction k Linear operator T −1 Reflection across line x1 =x2 Expansion or compression in x1 −direction 0 k −1 Expansion or compression in x2 −direction k 1 −k Shear in x1 −direction 1 −k Shear in x2 −direction Next, let us consider the question of elementary row operations on × matrices It is not difficult to see that an elementary row operation performed on a × matrix A has the effect of multiplying the Chapter : Linear Transformations page of 35 c Linear Algebra W W L Chen, 1997, 2008 matrix A by some elementary matrix E to give the product EA We have the following table Elementary row operation Elementary matrix E Interchanging the two rows 1 Multiplying row by non-zero factor k k 0 Multiplying row by non-zero factor k 0 k Adding k times row to row 1 k Adding k times row to row k Now, we know that any invertible matrix A can be reduced to the identity matrix by a finite number of elementary row operations In other words, there exist a finite number of elementary matrices E1 , , Es of the types above with various non-zero values of k such that Es E1 A = I, so that A = E1−1 Es−1 We have proved the following result PROPOSITION 8B Suppose that the linear operator T : R2 → R2 has standard matrix A, where A is invertible Then T is the product of a succession of finitely many reflections, expansions, compressions and shears In fact, we can prove the following result concerning images of straight lines PROPOSITION 8C Suppose that the linear operator T : R2 → R2 has standard matrix A, where A is invertible Then (a) the image under T of a straight line is a straight line; (b) the image under T of a straight line through the origin is a straight line through the origin; and (c) the images under T of parallel straight lines are parallel straight lines Proof Suppose that T (x1 , x2 ) = (y1 , y2 ) Since A is invertible, we have x = A−1 y, where x= x1 x2 and y= y1 y2 The equation of a straight line is given by αx1 + βx2 = γ or, in matrix form, by (α β) x1 x2 = (γ ) Hence (α Chapter : Linear Transformations β ) A−1 y1 y2 = (γ ) page of 35 c Linear Algebra W W L Chen, 1997, 2008 Let (α β ) = (α β ) A−1 Then (α β ) y1 y2 = (γ ) In other words, the image under T of the straight line αx1 + βx2 = γ is α y1 + β y2 = γ, clearly another straight line This proves (a) To prove (b), note that straight lines through the origin correspond to γ = To prove (c), note that parallel straight lines correspond to different values of γ for the same values of α and β 8.3 Elementary Properties of Euclidean Linear Transformations In this section, we establish a number of simple properties of euclidean linear transformations PROPOSITION 8D Suppose that T1 : Rn → Rm and T2 : Rm → Rk are linear transformations Then T = T2 ◦ T1 : Rn → Rk is also a linear transformation Proof Since T1 and T2 are linear transformations, they have standard matrices A1 and A2 respectively In other words, we have T1 (x) = A1 x for every x ∈ Rn and T2 (y) = A2 y for every y ∈ Rm It follows that T (x) = T2 (T1 (x)) = A2 A1 x for every x ∈ Rn , so that T has standard matrix A2 A1 Example 8.3.1 Suppose that T1 : R2 → R2 is anticlockwise rotation by π/2 and T2 : R2 → R2 is orthogonal projection onto the x1 -axis Then the respective standard matrices are A1 = −1 and A2 = 0 It follows that the standard matrices for T2 ◦ T1 and T1 ◦ T2 are respectively A2 A1 = 0 −1 and A1 A2 = 0 Hence T2 ◦ T1 and T1 ◦ T2 are not equal Example 8.3.2 Suppose that T1 : R2 → R2 is anticlockwise rotation by θ and T2 : R2 → R2 is anticlockwise rotation by φ Then the respective standard matrices are A1 = cos θ sin θ − sin θ cos θ and A2 = cos φ sin φ − sin φ cos φ It follows that the standard matrix for T2 ◦ T1 is A2 A1 = cos φ cos θ − sin φ sin θ sin φ cos θ + cos φ sin θ − cos φ sin θ − sin φ cos θ cos φ cos θ − sin φ sin θ = cos(φ + θ) − sin(φ + θ) sin(φ + θ) cos(φ + θ) Hence T2 ◦ T1 is anticlockwise rotation by φ + θ Example 8.3.3 The reader should check that in R2 , reflection across the x1 -axis followed by reflection across the x2 -axis gives reflection across the origin Linear transformations that map distinct vectors to distinct vectors are of special importance Chapter : Linear Transformations page of 35 c Linear Algebra W W L Chen, 1997, 2008 Definition A linear transformation T : Rn → Rm is said to be one-to-one if for every x , x ∈ Rn , we have x = x whenever T (x ) = T (x ) Example 8.3.4 If we consider linear operators T : R2 → R2 , then T is one-to-one precisely when the standard matrix A is invertible To see this, suppose first of all that A is invertible If T (x ) = T (x ), then Ax = Ax Multiplying on the left by A−1 , we obtain x = x Suppose next that A is not invertible Then there exists x ∈ R2 such that x = and Ax = On the other hand, we clearly have A0 = It follows that T (x) = T (0), so that T is not one-to-one PROPOSITION 8E Suppose that the linear operator T : Rn → Rn has standard matrix A Then the following statements are equivalent: (a) The matrix A is invertible (b) The linear operator T is one-to-one (c) The range of T is Rn ; in other words, R(T ) = Rn Proof ((a)⇒(b)) Suppose that T (x ) = T (x ) Then Ax = Ax Multiplying on the left by A−1 gives x =x ((b)⇒(a)) Suppose that T is one-to-one Then the system Ax = has unique solution x = in Rn It follows that A can be reduced by elementary row operations to the identity matrix I, and is therefore invertible ((a)⇒(c)) For any y ∈ Rn , clearly x = A−1 y satisfies Ax = y, so that T (x) = y ((c)⇒(a)) Suppose that {e1 , , en } is the standard basis for Rn Let x1 , , xn ∈ Rn be chosen to satisfy T (xj ) = ej , so that Axj = ej , for every j = 1, , n Write C = ( x1 xn ) Then AC = I, so that A is invertible Definition Suppose that the linear operator T : Rn → Rn has standard matrix A, where A is invertible Then the linear operator T −1 : Rn → Rn , defined by T −1 (x) = A−1 x for every x ∈ Rn , is called the inverse of the linear operator T Remark Clearly T −1 (T (x)) = x and T (T −1 (x)) = x for every x ∈ Rn Example 8.3.5 Consider the linear operator T : R2 → R2 , defined by T (x) = Ax for every x ∈ R2 , where A= 1 Clearly A is invertible, and A−1 = −1 −1 Hence the inverse linear operator is T −1 : R2 → R2 , defined by T −1 (x) = A−1 x for every x ∈ R2 Example 8.3.6 Suppose that T : R2 → R2 is anticlockwise rotation by angle θ The reader should check that T −1 : R2 → R2 is anticlockwise rotation by angle 2π − θ Next, we study the linearity properties of euclidean linear transformations which we shall use later to discuss linear transformations in arbitrary real vector spaces Chapter : Linear Transformations page of 35 c Linear Algebra W W L Chen, 1997, 2008 PROPOSITION 8F A transformation T : Rn → Rm is linear if and only if the following two conditions are satisfied: (a) For every u, v ∈ Rn , we have T (u + v) = T (u) + T (v) (b) For every u ∈ Rn and c ∈ R, we have T (cu) = cT (u) Proof Suppose first of all that T : Rn → Rm is a linear transformation Let A be the standard matrix for T Then for every u, v ∈ Rn and c ∈ R, we have T (u + v) = A(u + v) = Au + Av = T (u) + T (v) and T (cu) = A(cu) = c(Au) = cT (u) Suppose now that (a) and (b) hold To show that T is linear, we need to find a matrix A such that T (x) = Ax for every x ∈ Rn Suppose that {e1 , , en } is the standard basis for Rn As suggested by Proposition 8A, we write A = ( T (e1 ) T (en ) ) , where T (ej ) is a column matrix for every j = 1, , n For any vector   x1 x =   xn in Rn , we have  Ax = ( T (e1 )  x1 T (en ) )   = x1 T (e1 ) + + xn T (en ) xn Using (b) on each summand and then using (a) inductively, we obtain Ax = T (x1 e1 ) + + T (xn en ) = T (x1 e1 + + xn en ) = T (x) as required To conclude our study of euclidean linear transformations, we briefly mention the problem of eigenvalues and eigenvectors of euclidean linear operators Definition Suppose that T : Rn → Rn is a linear operator Then any real number λ ∈ R is called an eigenvalue of T if there exists a non-zero vector x ∈ Rn such that T (x) = λx This non-zero vector x ∈ Rn is called an eigenvector of T corresponding to the eigenvalue λ Remark Note that the equation T (x) = λx is equivalent to the equation Ax = λx It follows that there is no distinction between eigenvalues and eigenvectors of T and those of the standard matrix A We therefore not need to discuss this problem any further 8.4 General Linear Transformations Suppose that V and W are real vector spaces To define a linear transformation from V into W , we are motivated by Proposition 8F which describes the linearity properties of euclidean linear transformations Chapter : Linear Transformations page 10 of 35 c cc Linear Algebra Linear Algebra Linear Algebra W W L Chen, 1997, 2006 W W L Chen, 1997, 2006 2008 W W L Chen, 1997, 2006 Suppose next that = {w w } is basis of W Then we can define linear transformation Suppose next next that CC C= = {w {w11,,, ,,, w wm is aaa basis basis of of W W Then Then we we can can define define aaa linear linear transformation transformation m } is m that Suppose [w]C for m } every ψ : W → R w ∈ W , in a similar way We now have the following m , where ψ(w) = ψ : W → R m , where ψ(w) = [w]C for every w ∈ W , in a similar way We now have the following ψ : W →ofRlinear , where ψ(w) = [w]C for every w ∈ W , in a similar way We now have the following diagram transformations diagram of of linear linear transformations transformations diagram T T T V VV φ−1 φ−1 φ−1 ψ −1 ψ −1 ψ −1 φ φ φ n R Rnn R Clearly Clearly the the composition composition Clearly Clearly the the composition composition W W W ψ ψ ψ m R m Rm R −1 n m −1 n m S= = ψ T φ R → R −1 :: R m S ψ T φ → R =ψ ψ ◦◦◦◦ T T ◦◦◦◦ φ φ−1 Rnn → →R Rm SS = :: R is a euclidean linear transformation, and can therefore be described in terms of a standard matrix A is aa euclidean euclidean linear linear transformation, transformation, and and can therefore therefore be be described described in terms terms of of aa standard standard matrix matrix A A is Our task is to determine this matrix A in can terms of T and the bases Binand C Our task is to determine this matrix A in terms of T and the bases B and C Our task is to determine this matrix A in terms of T and the bases B and C We know from Proposition 8A that We know know from from Proposition Proposition 8A 8A that that We A = ( S(e1 ) S(en ) ) , A= = (( S(e S(e1)) S(e S(en)) )) ,, A n where {e1 , , en } is the standard basis for Rnn For every j = 1, , n, we have where {e {e1,, ,, een}} is is the the standard standard basis basis for for R Rn For For every every jj = = 1, 1, ,, n, n, we we have have where n −1 −1 S(ej ) = (ψ ◦ T ◦ φ −1)(ej ) = ψ(T (φ −1(ej ))) = ψ(T (vj )) = [T (vj )]C S(ej)) = = (ψ (ψ ◦◦ TT ◦◦ φφ−1 )(e )(ej)) = = ψ(T ψ(T(φ (φ−1 (e (ej))) ))) = = ψ(T ψ(T(v (vj)) )) = = [T [T(v (vj)] )]C S(e j j j j j C It follows that It follows follows that that It A = ( [T (v1 )]C [T (vn )]C ) (9) (9) )]C [T [T(v (vn)] )]C ) (9) A= = (( [T [T(v (v1)] (9) A C n C ) Definition The matrix A given by (9) is called the matrix for the linear transformation T with respect Definition The The matrix matrix A A given given by by (9) (9) is is called called the the matrix matrix for for the the linear linear transformation transformation T T with with respect respect Definition to the bases B and C to the bases B and C to the bases B and C We now have the following diagram of linear transformations We now now have have the the following following diagram diagram of of linear linear transformations transformations We T T T V VV φ−1 φ−1 φ−1 W W W ψ −1 ψ −1 ψ −1 φ φ φ ψ ψ ψ S n m S R R m S Rnn Rm R R Hence we we can can write write T T as as the the composition composition Hence Hence we can write T as the composition −1 T= =ψ ψ−1 ◦ S ◦ φ : V → W T −1 ◦ S ◦ φ : V → W T =ψ ◦ S ◦ φ : V → W For every v ∈ V , we have the following: For every every v v∈ ∈V V ,, we we have have the the following: following: For v vv φ φ φ Chapter : Linear Transformations Chapter : Linear Transformations Chapter : Linear Transformations [v] [v]B B [v] B S S S A[v] A[v]B B A[v] B ψ −1 ψ −1 ψ −1 −1 ψ −1 (A[v]B ) ψ−1 (A[v]B)) ψ (A[v] B page 21 of 35 page 21 of 35 page 21 of 35 c Linear Algebra W W L Chen, 1997, 2008 More precisely, if v = β1 v1 + + βn , then   β1   [v]B =    and βn   β1    A[v]B = A   =  βn  γ1  ,  γm say, and so T (v) = ψ −1 (A[v]B ) = γ1 w1 + + γm wm We have proved the following result PROPOSITION 8S Suppose that T : V → W is a linear transformation from a real vector space V into a real vector space W Suppose further that V and W are finite dimensional, with bases B and C respectively, and that A is the matrix for the linear transformation T with respect to the bases B and C Then for every v ∈ V , we have T (v) = w, where w ∈ W is the unique vector satisfying [w]C = A[v]B Remark In the special case when V = W , the linear transformation T : V → W is a linear operator on T Of course, we may choose a basis B for the domain V of T and a basis C for the codomain V of T In the case when T is the identity linear operator, we often choose B = C since this represents a change of basis In the case when T is not the identity operator, we often choose B = C for the sake of convenience; we then say that A is the matrix for the linear operator T with respect to the basis B Example 8.8.1 Consider an operator T : P3 → P3 on the real vector space P3 of all polynomials with real coefficients and degree at most 3, where for every polynomial p(x) in P3 , we have T (p(x)) = xp (x), the product of x with the formal derivative p (x) of p(x) The reader is invited to check that T is a linear operator Now consider the basis B = {1, x, x2 , x3 } of P3 The matrix for T with respect to B is given by  A = ( [T (1)]B [T (x)]B [T (x2 )]B [T (x3 )]B ) = ( [0]B [x]B [2x2 ]B 0 [3x ]B ) =  0 0 0  0  Suppose that p(x) = + 2x + 4x2 + 3x3 Then   2 [p(x)]B =    and 0 A[p(x)]B =  0 0 0     02 2   =  , 3 so that T (p(x)) = 2x + 8x2 + 9x3 This can be easily verified by noting that T (p(x)) = xp (x) = x(2 + 8x + 9x2 ) = 2x + 8x2 + 9x3 In general, if p(x) = p0 + p1 x + p2 x2 + p3 x3 , then   p0 p  [p(x)]B =   p2 p3  and 0 A[p(x)]B =  0 0 0     p0 0   p1   p1    =  , p2 2p2 p3 3p3 so that T (p(x)) = p1 x + 2p2 x2 + 3p3 x3 Observe that T (p(x)) = xp (x) = x(p1 + 2p2 x + 3p3 x2 ) = p1 x + 2p2 x2 + 3p3 x3 , verifying our result Chapter : Linear Transformations page 22 of 35 c Linear Linear Algebra Algebra W WW WL L Chen, Chen, 1997, 1997, 2006 2008 Example 8.8.2 Consider the linear operator T : R22 → R22, given by T (x11, x22) = (2x11 + x22, x11 + 3x22) for every (x11, x22) ∈ R22 Consider also the basis B = {(1, 0), (1, 1)} of R22 Then the matrix for T with respect to B is given by −1 A = ( [T (1, 0)]BB [T (1, 1)]BB ) = ( [(2, 1)]BB [(3, 4)]BB ) = Suppose that (x11, x22) = (3, 2) Then [(3, 2)]BB = and A[(3, 2)]BB = −1 = −1 , so that T (3, 2) = −(1, 0) + 9(1, 1) = (8, 9) This can be easily verified directly In general, we have [(x11, x22)]BB = x11 − x22 x22 and A[(x11, x22)]BB = −1 x11 − x22 x22 = x11 − 2x22 x11 + 3x22 , so that T (x11, x22) = (x11 − 2x22)(1, 0) + (x11 + 3x22)(1, 1) = (2x11 + x22, x11 + 3x22) m Example 8.8.3 Suppose that T : Rnn → Rm is a linear transformation Suppose further that B and C n m n m are the standard bases for R and R respectively Then the matrix for T with respect to B and C is given by A = ( [T (e11)]CC [T (enn)]CC ) = ( T (e11) T (enn) ) , so it follows from Proposition 8A that A is simply the standard matrix for T Suppose now that T11 : V → W and T22 : W → U are linear transformations, where the real vector spaces V, W, U are finite dimensional, with respective bases B = {v11, , vnn}, C = {w11, , wm m} and D = {u11, , ukk} We then have the following diagram of linear transformations T1 V φ−1 ψ −1 φ Rn S1 T2 W η −1 ψ Rm U S2 η Rk Here η : U → Rkk, where η(u) = [u]D D for every u ∈ U , is a linear transformation, and −1 : Rn n → Rm m S11 = ψ ◦ T11 ◦ φ−1 and −1 : Rm m → Rkk S22 = η ◦ T22 ◦ ψ −1 are euclidean linear transformations Suppose that A11 and A22 are respectively the standard matrices for S11 and S22, so that they are respectively the matrix for T11 with respect to B and C and the matrix for T22 with respect to C and D Clearly −1 : Rn n → Rkk S22 ◦ S11 = η ◦ T22 ◦ T11 ◦ φ−1 It follows that A22A11 is the standard matrix for S22 ◦ S11, and so is the matrix for T22 ◦ T11 with respect to the bases B and D To summarize, we have the following result Chapter Chapter 88 :: Linear Linear Transformations Transformations page page 23 23 of of 35 35 c Linear Algebra W W L Chen, 1997, 2008 PROPOSITION 8T Suppose that T1 : V → W and T2 : W → U are linear transformations, where the real vector spaces V, W, U are finite dimensional, with bases B, C, D respectively Suppose further that A1 is the matrix for the linear transformation T1 with respect to the bases B and C, and that A2 is the matrix for the linear transformation T2 with respect to the bases C and D Then A2 A1 is the matrix for the linear transformation T2 ◦ T1 with respect to the bases B and D Example 8.8.4 Consider the linear operator T1 : P3 → P3 , where for every polynomial p(x) in P3 , we have T1 (p(x)) = xp (x) We have already shown that the matrix for T1 with respect to the basis B = {1, x, x2 , x3 } of P3 is given by  0 A1 =  0 0 0  0  Consider next the linear operator T2 : P3 → P3 , where for every polynomial q(x) = q0 + q1 x + q2 x2 + q3 x3 in P3 , we have T2 (q(x)) = q(1 + x) = q0 + q1 (1 + x) + q2 (1 + x)2 + q3 (1 + x)3 We have T2 (1) = 1, T2 (x) = + x, T2 (x2 ) = + 2x + x2 and T2 (x3 ) = + 3x + 3x2 + x3 , so that the matrix for T2 with respect to B is given by  A2 = ( [T2 (1)]B [T2 (x)]B 0 [T2 (x )]B ) =  0 [T2 (x2 )]B 1 0  3  Consider now the composition T = T2 ◦ T1 : P3 → P3 Let A denote the matrix for T with respect to B By Proposition 8T, we have  0 A = A2 A1 =  0 1 0  30  0 0   0 0 0 = 0 0 1 0  9  Suppose that p(x) = p0 + p1 x + p2 x2 + p3 x3 Then   p0 p  [p(x)]B =   p2 p3  and 0 A[p(x)]B =  0 1 0     p0 p1 + 2p2 + 3p3   p1   p1 + 4p2 + 9p3    =  , p2 2p2 + 9p3 p3 3p3 so that T (p(x)) = (p1 + 2p2 + 3p3 ) + (p1 + 4p2 + 9p3 )x + (2p2 + 9p3 )x2 + 3p3 x3 We can check this directly by noting that T (p(x)) = T2 (T1 (p(x))) = T2 (p1 x + 2p2 x2 + 3p3 x3 ) = p1 (1 + x) + 2p2 (1 + x)2 + 3p3 (1 + x)3 = (p1 + 2p2 + 3p3 ) + (p1 + 4p2 + 9p3 )x + (2p2 + 9p3 )x2 + 3p3 x3 Example 8.8.5 Consider the linear operator T : R2 → R2 , given by T (x1 , x2 ) = (2x1 + x2 , x1 + 3x2 ) for every (x1 , x2 ) ∈ R2 We have already shown that the matrix for T with respect to the basis B = {(1, 0), (1, 1)} of R2 is given by A= Chapter : Linear Transformations 1 −1 page 24 of 35 c Linear Algebra W W L Chen, 1997, 2008 Consider the linear operator T : R2 → R2 By Proposition 8T, the matrix for T with respect to B is given by 1 A2 = −1 1 −1 = −5 15 Suppose that (x1 , x2 ) ∈ R2 Then [(x1 , x2 )]B = x1 − x2 x2 A2 [(x1 , x2 )]B = and −5 15 x1 − x2 x2 = −5x2 5x1 + 10x2 , so that T (x1 , x2 ) = −5x2 (1, 0) + (5x1 + 10x2 )(1, 1) = (5x1 + 5x2 , 5x1 + 10x2 ) The reader is invited to check this directly A simple consequence of Propositions 8N and 8T is the following result concerning inverse linear transformations PROPOSITION 8U Suppose that T : V → V is a linear operator on a finite dimensional real vector space V with basis B Suppose further that A is the matrix for the linear operator T with respect to the basis B Then T is one-to-one if and only if A is invertible Furthermore, if T is one-to-one, then A−1 is the matrix for the inverse linear operator T −1 : V → V with respect to the basis B Proof Simply note that T is one-to-one if and only if the system Ax = has only the trivial solution x = The last assertion follows easily from Proposition 8T, since if A denotes the matrix for the inverse linear operator T −1 with respect to B, then we must have A A = I, the matrix for the identity operator T −1 ◦ T with respect to B Example 8.8.6 Consider the linear operator T : P3 → P3 , where for every q(x) = q0 + q1 x + q2 x2 + q3 x3 in P3 , we have T (q(x)) = q(1 + x) = q0 + q1 (1 + x) + q2 (1 + x)2 + q3 (1 + x)3 We have already shown that the matrix for T with respect to the basis B = {1, x, x2 , x3 } is given by  0 A= 0 1 0  3  This matrix is invertible, so it follows that T is one-to-one Furthermore, it can be checked that  A−1 0 = 0 −1 0  −1 −2   −3 Suppose that p(x) = p0 + p1 x + p2 x2 + p3 x3 Then   p0 p  [p(x)]B =   p2 p3  and  A−1 [p(x)]B =  0 −1 0 −2     −1 p0 p0 − p1 + p2 − p3   p1   p1 − 2p2 + 3p3    =  , −3 p2 p2 − 3p3 p3 p3 so that T −1 (p(x)) = (p0 − p1 + p2 − p3 ) + (p1 − 2p2 + 3p3 )x + (p2 − 3p3 )x2 + p3 x3 = p0 + p1 (x − 1) + p2 (x2 − 2x + 1) + p3 (x3 − 3x2 + 3x − 1) = p0 + p1 (x − 1) + p2 (x − 1)2 + p3 (x − 1)3 = p(x − 1) Chapter : Linear Transformations page 25 of 35 c Linear Algebra W W L Chen, 1997, 2008 8.9 Change of Basis Suppose that V is a finite dimensional real vector space, with one basis B = {v1 , , } and another basis B = {u1 , , un } Suppose that T : V → V is a linear operator on V Let A denote the matrix for T with respect to the basis B, and let A denote the matrix for T with respect to the basis B If v ∈ V and T (v) = w, then [w]B = A[v]B (10) [w]B = A [v]B (11) and We wish to find the relationship between A and A Recall Proposition 8J, that if P = ( [u1 ]B [un ]B ) denotes the transition matrix from the basis B to the basis B, then [v]B = P [v]B and [w]B = P [w]B (12) Note that the matrix P can also be interpreted as the matrix for the identity operator I : V → V with respect to the bases B and B It is easy to see that the matrix P is invertible, and P −1 = ( [v1 ]B [vn ]B ) denotes the transition matrix from the basis B to the basis B , and can also be interpreted as the matrix for the identity operator I : V → V with respect to the bases B and B Combining (10) and (12), we conclude that [w]B = P −1 [w]B = P −1 A[v]B = P −1 AP [v]B Comparing this with (11), we conclude that P −1 AP = A (13) A = P A P −1 (14) This implies that Remark We can use the notation A = [T ]B and A = [T ]B to denote that A and A are the matrices for T with respect to the basis B and with respect to the basis B respectively We can also write P = [I]B,B to denote that P is the transition matrix from the basis B to the basis B, so that P −1 = [I]B ,B Chapter : Linear Transformations page 26 of 35 cc Linear Linear Algebra Algebra W WW WL L Chen, Chen, 1997, 1997, 2006 2008 Then Then (13) (13) and and (14) (14) become become respectively respectively [I] [T ]B [I] = [T ]B [I]B [I]B,B B ,B ,B [T ]B B,B = [T ]B and and [I] [T ]B [I] = [T ]B [I]B,B [I]B B,B [T ]B B ,B ,B = [T ]B We have proved the following result PROPOSITION 8V Suppose that T : V → V is a linear operator on a finite dimensional space V , with bases B = {v11 , , vnn } and B = {u11 , , unn } Suppose further that A and matrices for T with respect to the basis B and with respect to the basis B respectively Then P −1 AP = A real vector A are the A = P AP −1 , and where P = ( [u1 ]B [un ]B ) denotes the transition matrix from the basis B to the basis B Remarks (1) We have the following picture T v w I I v w T A [v]B [w]B P P −1 [v]B A [w]B (2) (2) The The idea idea can can be be extended extended to to the the case case of of linear linear transformations transformations T T :: V V → →W W from from aa finite finite dimensional dimensional real vector space into another, with a change of basis in V and a change of basis in W real vector space into another, with a change of basis in V and a change of basis in W Example Example 8.9.1 8.9.1 Consider Consider the the vector vector space space P P33 of of all all polynomials polynomials with with real real coefficients coefficients and and degree degree at at 2 , x3 } and B = {1, + x, + x + x2 , + x + x2 + x3 } Consider also most 3, with bases B = {1, x, x most 3, with bases B = {1, x, x , x } and B = {1, + x, + x + x , + x + x + x } Consider also the the linear linear operator operator T T :: P P33 → →P P33 ,, where where for for every every polynomial polynomial p(x) p(x) = = pp00 + + pp11 x x+ + pp22 x x2 + + pp33 x x3 ,, we we have have + (p0 + p3 )x3 Let A denote the matrix for T with respect T (p(x)) = (p + p ) + (p + p )x + (p + p )x 1 2 T (p(x)) = (p0 + p1 ) + (p1 + p2 )x + (p2 + p3 )x + (p0 + p3 )x Let A denote the matrix for T with respect 2 3 to to the the basis basis B B Then Then T T (1) (1) = = 11 + +x x3 ,, T T (x) (x) = = 11 + + x, x, T T (x (x2 )) = =x x+ +x x2 and and T T (x (x3 )) = =x x2 + +x x3 ,, and and so so A A= = (( [T [T (1)] (1)]B B [T [T (x)] (x)]B B [T [T (x (x2 )] )]B B  1 00   [T )= [T (x (x3 )] )]B B ) = 0 11 11 11 00 00 00 11 11 00  00  00    11  11 Next, Next, note note that that the the transition transition matrix matrix from from the the basis basis B B to to the the basis basis B B is is given given by by  1 1  2 11 P [1 [1 [1 )= P = = (( [1] [1]B [1 + + x] x]B [1 + +x x+ +x x2 ]]B [1 + +x x+ +x x2 + +x x3 ]]B 0 B B B B)= 0 00 00 Chapter Chapter 8 :: Linear Linear Transformations Transformations 11 11 11 00  11  11    11  11 page page 27 27 of of 35 35 c Linear Algebra W W L Chen, 1997, 2008 It can be checked that  P −1 0 = 0 −1 0 −1  0  , −1 and so  0 −1 A = P AP =  0 −1 0 −1  0  −1 1 1 0 1  00  1 1 0 1   1 1  = −1 1 1 −1  0 0  0 is the matrix for T with respect to the basis B It follows that T (1) = − (1 + x + x2 ) + (1 + x + x2 + x3 ) = + x3 , T (1 + x) = + (1 + x) − (1 + x + x2 ) + (1 + x + x2 + x3 ) = + x + x3 , T (1 + x + x2 ) = (1 + x) + (1 + x + x2 + x3 ) = + 2x + x2 + x3 , T (1 + x + x2 + x3 ) = 2(1 + x + x2 + x3 ) = + 2x + 2x2 + 2x3 These can be verified directly 8.10 Eigenvalues and Eigenvectors Definition Suppose that T : V → V is a linear operator on a finite dimensional real vector space V Then any real number λ ∈ R is called an eigenvalue of T if there exists a non-zero vector v ∈ V such that T (v) = λv This non-zero vector v ∈ V is called an eigenvector of T corresponding to the eigenvalue λ The purpose of this section is to show that the problem of eigenvalues and eigenvectors of the linear operator T can be reduced to the problem of eigenvalues and eigenvectors of the matrix for T with respect to any basis B of V The starting point of our argument is the following theorem, the proof of which is left as an exercise PROPOSITION 8W Suppose that T : V → V is a linear operator on a finite dimensional real vector space V , with bases B and B Suppose further that A and A are the matrices for T with respect to the basis B and with respect to the basis B respectively Then (a) det A = det A ; (b) A and A have the same rank; (c) A and A have the same characteristic polynomial; (d) A and A have the same eigenvalues; and (e) the dimension of the eigenspace of A corresponding to an eigenvalue λ is equal to the dimension of the eigenspace of A corresponding to λ We also state without proof the following result PROPOSITION 8X Suppose that T : V → V is a linear operator on a finite dimensional real vector space V Suppose further that A is the matrix for T with respect to a basis B of V Then (a) the eigenvalues of T are precisely the eigenvalues of A; and (b) a vector u ∈ V is an eigenvector of T corresponding to an eigenvalue λ if and only if the coordinate matrix [u]B is an eigenvector of A corresponding to the eigenvalue λ Chapter : Linear Transformations page 28 of 35 c Linear Algebra W W L Chen, 1997, 2008 Suppose now that A is the matrix for a linear operator T : V → V on a finite dimensional real vector space V with respect to a basis B = {v1 , , } If A can be diagonalized, then there exists an invertible matrix P such that P −1 AP = D is a diagonal matrix Furthermore, the columns of P are eigenvectors of A, and so are the coordinate matrices of eigenvectors of T with respect to the basis B In other words, P = ( [u1 ]B [un ]B ) , where B = {u1 , , un } is a basis of V consiting of eigenvectors of T Furthermore, P is the transition matrix from the basis B to the basis B It follows that the matrix for T with respect to the basis B is given by  D=  λ1 , λn where λ1 , , λn are the eigenvalues of T Example 8.10.1 Consider the vector space P2 of all polynomials with real coefficients and degree at most 2, with basis B = {1, x, x2 } Consider also the linear operator T : P2 → P2 , where for every polynomial p(x) = p0 + p1 x + p2 x2 , we have T (p(x)) = (5p0 − 2p1 ) + (6p1 + 2p2 − 2p0 )x + (2p1 + 7p2 )x2 Then T (1) = − 2x, T (x) = −2 + 6x + 2x2 and T (x2 ) = 2x + 7x2 , so that the matrix for T with respect to the basis B is given by  A = ( [T (1)]B [T (x)]B [T (x2 )]B ) =  −2 −2  2 It is a simple exercise to show that the matrix A has eigenvalues 3, 6, 9, with corresponding eigenvectors   x1 =   , −1  −1 x3 =   ,   x2 =  −1  ,  so that writing  P = −1  −1 , 2 −1 we have  P −1 AP =  0  0 Now let B = {p1 (x), p2 (x), p3 (x)}, where   [p1 (x)]B =   , −1 Chapter : Linear Transformations   [p2 (x)]B =  −1  ,   −1 [p3 (x)]B =   page 29 of 35 c Linear Algebra W W L Chen, 1997, 2008 Then P is the transition matrix from the basis B to the basis B, and D is the matrix for T with respect to the basis B Clearly p1 (x) = + 2x − x2 , p2 (x) = − x + 2x2 and p3 (x) = −1 + 2x + 2x2 Note now that T (p1 (x)) = T (2 + 2x − x2 ) = + 6x − 3x2 = 3p1 (x), T (p2 (x)) = T (2 − x + 2x2 ) = 12 − 6x + 12x2 = 6p2 (x), T (p3 (x)) = T (−1 + 2x + 2x2 ) = −9 + 18x + 18x2 = 9p3 (x) Chapter : Linear Transformations page 30 of 35 c Linear Algebra W W L Chen, 1997, 2008 Problems for Chapter Consider the transformation T : R3 → R4 , given by T (x1 , x2 , x3 ) = (x1 + x2 + x3 , x2 + x3 , 3x1 + x2 , 2x2 + x3 ) for every (x1 , x2 , x3 ) ∈ R3 a) Find the standard matrix A for T b) By reducing A to row echelon form, determine the dimension of the kernel of T and the dimension of the range of T Consider a linear operator T : R3 → R3 with standard matrix  A = 2  3 2 Let {e1 , e2 , e3 } denote the standard basis for R3 a) Find T (ej ) for every j = 1, 2, b) Find T (2e1 + 5e2 + 3e3 ) c) Is T invertible? Justify your assertion Consider the linear operator T : R2 → R2 with standard matrix A= 1 a) Find the image under T of the line x1 + 2x2 = b) Find the image under T of the circle x21 + x22 = For each of the following, determine whether the given transformation is linear: a) T : V → R, where V is a real inner product space and T (u) = u b) T : M2,2 (R) → M2,3 (R), where B ∈ M2,3 (R) is fixed and T (A) = AB c) T : M3,4 (R) → M4,3 (R), where T (A) = At d) T : P2 → P2 , where T (p0 + p1 x + p2 x2 ) = p0 + p1 (2 + x) + p2 (2 + x)2 e) T : P2 → P2 , where T (p0 + p1 x + p2 x2 ) = p0 + p1 x + (p2 + 1)x2 Suppose that T : R3 → R3 is a linear transformation satisfying the conditions T (1, 0, 0) = (2, 4, 1), T (1, 1, 0) = (3, 0, 2) and T (1, 1, 1) = (1, 4, 6) a) Evaluate T (5, 3, 2) b) Find T (x1 , x2 , x3 ) for every (x1 , x2 , x3 ) ∈ R3 Suppose that T : R3 → R3 is orthogonal projection onto the x1 x2 -plane a) Find the standard matrix A for T b) Find A2 c) Show that T ◦ T = T Consider the bases B = {u1 , u2 , u3 } and C = {v1 , v2 , v3 } of R3 , where u1 = (2, 1, 1), u2 = (2, −1, 1), u3 = (1, 2, 1), v1 = (3, 1, −5), v2 = (1, 1, −3) and v3 = (−1, 0, 2) a) Find the transition matrix from the basis C to the basis B b) Find the transition matrix from the basis B to the basis C c) Show that the matrices in parts (a) and (b) are inverses of each other d) Compute the coordinate matrix [u]C , where u = (−5, 8, −5) e) Use the transition matrix to compute the coordinate matrix [u]B f) Compute the coordinate matrix [u]B directly and compare it to your answer in part (e) Chapter : Linear Transformations page 31 of 35 c Linear Algebra W W L Chen, 1997, 2008 Consider the bases B = {p1 , p2 } and C = {q1 , q2 } of P1 , where p1 = 2, p2 = + 2x, q1 = + 3x and q2 = 10 + 2x a) Find the transition matrix from the basis C to the basis B b) Find the transition matrix from the basis B to the basis C c) Show that the matrices in parts (a) and (b) are inverses of each other d) Compute the coordinate matrix [p]C , where p = −4 + x e) Use the transition matrix to compute the coordinate matrix [p]B f) Compute the coordinate matrix [p]B directly and compare it to your answer in part (e) Let V be the real vector space spanned by the functions f1 = sin x and f2 = cos x a) Show that g1 = sin x + cos x and g2 = cos x form a basis of V b) Find the transition matrix from the basis C = {g1 , g2 } to the basis B = {f1 , f2 } of V c) Compute the coordinate matrix [f ]C , where f = sin x − cos x d) Use the transition matrix to compute the coordinate matrix [f ]B e) Compute the coordinate matrix [f ]B directly and compare it to your answer in part (d) 10 Let P be the transition matrix from a basis C to another basis B of a real vector space V Explain why P is invertible 11 For each of the following linear transformations T , find ker(T ) and R(T ), and verify the Rank-nullity theorem:   −1 a) T : R3 → R3 , with standard matrix A =  −4  b) T : P3 → P2 , where T (p(x)) = p (x), the formal derivative c) T : P1 → R, where T (p(x)) = p(x) dx 12 For each of the following, determine whether the linear operator T : Rn → Rn is one-to-one If so, find also the inverse linear operator T −1 : Rn → Rn : a) T (x1 , x2 , x3 , , xn ) = (x2 , x1 , x3 , , xn ) b) T (x1 , x2 , x3 , , xn ) = (x2 , x3 , , xn , x1 ) c) T (x1 , x2 , x3 , , xn ) = (x2 , x2 , x3 , , xn ) 13 Consider the operator T : R2 → R2 , where T (x1 , x2 ) = (x1 + kx2 , −x2 ) for every (x1 , x2 ) ∈ R2 Here k ∈ R is fixed a) Show that T is a linear operator b) Show that T is one-to-one c) Find the inverse linear operator T −1 : R2 → R2 14 Consider the linear transformation T : P2 → P1 , where T (p0 + p1 x + p2 x2 ) = (p0 + p2 ) + (2p0 + p1 )x for every polynomial p0 + p1 x + p2 x2 in P2 a) Find the matrix for T with respect to the bases {1, x, x2 } and {1, x} b) Find T (2 + 3x + 4x2 ) by using the matrix A c) Use the matrix A to recover the formula T (p0 + p1 x + p2 x2 ) = (p0 + p2 ) + (2p0 + p1 )x 15 Consider the linear operator T : R2 → R2 , where T (x1 , x2 ) = (x1 −x2 , x1 +x2 ) for every (x1 , x2 ) ∈ R2 a) Find the matrix A for T with respect to the basis {(1, 1), (−1, 0)} of R2 b) Use the matrix A to recover the formula T (x1 , x2 ) = (x1 − x2 , x1 + x2 ) c) Is T one-to-one? If so, use the matrix A to find the inverse linear operator T −1 : R2 → R2 Chapter : Linear Transformations page 32 of 35 c Linear Algebra W W L Chen, 1997, 2008 16 Consider the real vector space of all real sequences x = (x1 , x2 , x3 , ) such that the series ∞ xn n=1 is convergent a) Show that the transformation T : V → R, given by ∞ T (x) = xn n=1 for every x ∈ V , is a linear transformation b) Is the linear transformation T one-to-one? If so, give a proof If not, find two distinct vectors x, y ∈ V such that T (x) = T (y) 17 Suppose that T1 : R2 → R2 and T2 : R2 → R2 are linear operators such that T1 (x1 , x2 ) = (x1 + x2 , x1 − x2 ) and T2 (x1 , x2 ) = (2x1 + x2 , x1 − 2x2 ) for every (x1 , x2 ) ∈ R2 a) Show that T1 and T2 are one-to-one b) Find the formulas for T1−1 , T2−1 and (T2 ◦ T1 )−1 c) Verify that (T2 ◦ T1 )−1 = T1−1 ◦ T2−1 18 Consider the transformation T : P1 → R2 , where T (p(x)) = (p(0), p(1)) for every polynomial p(x) in P1 a) Find T (1 − 2x) b) Show that T is a linear transformation c) Show that T is one-to-one d) Find T −1 (2, 3), and sketch its graph 19 Suppose that V and W are finite dimensional real vector spaces with dim V > dim W Suppose further that T : V → W is a linear transformation Explain why T cannot be one-to-one 20 Suppose that  A = 2 −2  −1  is the matrix for a linear operator T : P2 → P2 with respect to the basis B = {p1 (x), p2 (x), p3 (x)} of P2 , where p1 (x) = 3x + 3x2 , p2 (x) = −1 + 3x + 2x2 and p3 (x) = + 7x + 2x2 a) Find [T (p1 (x))]B , [T (p2 (x))]B and [T (p3 (x))]B b) Find T (p1 (x)), T (p2 (x)) and T (p3 (x)) c) Find a formula for T (p0 + p1 x + p2 x2 ) d) Use the formula in part (c) to compute T (1 + x2 ) 21 Suppose that B = {v1 , v2 , v3 , v4 } is a basis for a real vector space V Suppose that T : V → V is a linear operator, with T (v1 ) = v2 , T (v2 ) = v4 , T (v3 ) = v1 and T (v4 ) = v3 a) Find the matrix for T with respect to the basis B b) Is T one-to-one? If so, describe its inverse Chapter : Linear Transformations page 33 of 35 c Linear Algebra W W L Chen, 1997, 2008 22 Let Pk denote the vector space of all polynomials with real coefficients and degree at most k Consider P2 with basis B = {1, x, x2 } and P3 with basis C = {1, x, x2 , x3 } We define T1 : P2 → P3 and T2 : P3 → P2 as follows For every polynomial p(x) = a0 + a1 x + a2 x2 in P2 , we have T1 (p(x)) = xp(x) = a0 x + a1 x2 + a2 x3 For every polynomial q(x) in P3 , we have T2 (q(x)) = q (x), the formal derivative of q(x) with respect to the variable x a) Show that T1 : P2 → P3 and T2 : P3 → P2 are linear transformations b) Find T1 (1), T1 (x), T1 (x2 ), and compute the matrix A1 for T1 : P2 → P3 with respect to the bases B and C c) Find T2 (1), T2 (x), T2 (x2 ), T2 (x3 ), and compute the matrix A2 for T2 : P3 → P2 with respect to the bases C and B d) Let T = T2 ◦ T1 Find T (1), T (x), T (x2 ), and compute the matrix A for T : P2 → P2 with respect to the basis B Verify that A = A2 A1 23 Suppose that T : V → V is a linear operator on a real vector space V with basis B Suppose that for every v ∈ V , we have     x1 − x2 + x3 x1 [T (v)]B =  x1 + x2  and [v]B =  x2  x1 − x2 x3 a) Find the matrix for T with respect to the basis B b) Is T one-to-one? If so, describe its inverse 24 For each of the following, let V be the subspace with basis B = {f1 (x), f2 (x), f3 (x)} of the space of all real valued functions defined on R Let T : V → V be defined by T (f (x)) = f (x) for every function f (x) in V Find the matrix for T with respect to the basis B: a) f1 (x) = 1, f2 (x) = sin x, f3 (x) = cos x b) f1 (x) = e2x , f2 (x) = xe2x , f3 (x) = x2 e2x 25 Let P2 denote the vector space of all polynomials with real coefficients and degree at most 2, with basis B = {1, x, x2 } Consider the linear operator T : P2 → P2 , where for every polynomial p(x) = a0 + a1 x + a2 x2 in P2 , we have T (p(x)) = p(2x + 1) = a0 + a1 (2x + 1) + a2 (2x + 1)2 a) Find T (1), T (x), T (x2 ), and compute the matrix A for T with respect to the basis B b) Use the matrix A to compute T (3 + x + 2x2 ) c) Check your calculations in part (b) by computing T (3 + x + 2x2 ) directly d) What is the matrix for T ◦ T : P2 → P2 with respect to the basis B? e) Consider a new basis B = {1 + x, + x2 , x + x2 } of P2 Using a change of basis matrix, compute the matrix for T with respect to the basis B f) Check your answer in part (e) by computing the matrix directly 26 Consider the linear operator T : P1 → P1 , where for every polynomial p(x) = p0 + p1 x in P1 , we have T (p(x)) = p0 + p1 (x + 1) Consider also the bases B = {6 + 3x, 10 + 2x} and B = {2, + 2x} of P1 a) Find the matrix for T with respect to the basis B b) Use Proposition 8V to compute the matrix for T with respect to the basis B 27 Suppose that V and W are finite dimensional real vector spaces Suppose further that B and B are bases for V , and that C and C are bases for W Show that for any linear transformation T : V → W , we have [I]C ,C [T ]C,B [I]B,B = [T ]C ,B 28 Prove Proposition 8W 29 Prove Proposition 8X Chapter : Linear Transformations page 34 of 35 c Linear Algebra W W L Chen, 1997, 2008 30 For each of the following linear transformations T : R3 → R3 , find a basis B of R3 such that the matrix for T with respect to the basis B is a diagonal matrix: a) T (x1 , x2 , x3 ) = (−x2 + x3 , −x1 + x3 , x1 + x2 ) b) T (x1 , x2 , x3 ) = (4x1 + x3 , 2x1 + 3x2 + 2x3 , x1 + 4x3 ) 31 Consider the linear operator T : P2 → P2 , where T (p0 + p1 x + p2 x2 ) = (p0 − 6p1 + 12p2 ) + (13p1 − 30p2 )x + (9p1 − 20p2 )x2 a) Find the eigenvalues of T b) Find a basis B of P2 such that the matrix for T with respect to B is a diagonal matrix Chapter : Linear Transformations page 35 of 35 ... following result Chapter Chapter 88 :: Linear Linear Transformations Transformations page page 23 23 of of 35 35 c Linear Algebra W W L Chen, 1997, 20 08 PROPOSITION 8T Suppose that T1 : V → W and... established by Proposition 8M The equivalence of (b) and (c) follows from Proposition 8L Chapter : Linear Transformations page 19 of 35 c Linear Algebra W W L Chen, 1997, 20 08 Suppose that T : V →... matrix A in terms of T and the bases B and C We know from Proposition 8A that We know know from from Proposition Proposition 8A 8A that that We A = ( S(e1 ) S(en ) ) , A= = (( S(e S(e1)) S(e