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Lecturer: Trần Thị Huê The textbook: Elementary Linear Algebra (posted on E_learning) This course is credits (45 periods) quizzes (10 – 15 mins), homeworks (5% for each) and final test (90mins, 50%) What is Linear Algebra? Suppose you travel on an airplane between two cities that are 5000 kms apart If the trip one way against a headwind takes 25/4 hours and the return trip the same day in the direction of the wind takes only hours, can you find the ground speed of the plane and the speed of the wind, assuming that both remain constant? Chapter 1: Systems of Linear Equations, Matrices, and determinants Chapter 2: Vector Spaces and Euclidean Spaces Chapter 3: Linear Transformations Chapter 4: Eigenvalues and Eigenvectors Chapter 1: Systems of Linear Equations, Matrices, and determinants 1.1 Introduction to Systems of Linear Equations 1.2 Matrices 1.3 Determinants 1.1 Introduction to Systems of Linear Equations Linear Equations in n variables A linear equation in n variables x1, x2, …, xn has the form a1 x1 + a2 x2 + ⋯ an xn = b (i = 1, , n), b ∈ R Linear equations have no products or roots of variables and no variables involved in trigonometric, exponential, or logarithmic functions Variables appear only to the first power Example: Each following equation is linear or not linear 1) x1 + x2 − 10 x3 + x4 = 10 2) x + y − π z = 2 3) xy + z = 4) e x − y = A solution of LEs in n variables is a sequence of n real numbers s1, s2, …, sn arranged so the equation is satisfied when those values are substituted into that equation Systems of Linear Equations A system of LEs in n variables is a set of m equations, each of which is linear in the same n variables a11 x1 + a12 x2 + ⋯ + a1n xn = b1 a x + a x + ⋯ + a x = b 21 22 2n n ⋯⋯⋯ am1 x1 + am x2 + ⋯ + amn xn = bm A solution of a system of LEs is a sequence of numbers s1, s2, …, sn that is a solution of each of the LE in the system A system of LEs is called consistent if it has at least one solution and inconsistent if it has no solution Example: solve each system of LEs x + y = 1) x − y = −1 x + y = 2) 2 x + y = x + y = 3) x + y = Solving a system of Linear Equations Which system is easier to solve algebraically? x − y + 3z = 1) − x + y = −4 2 x − y + z = 17 x − y + 3z = 2) y + 3z = z=2 Exp1) Let B={u11, 0), u2(0, 1)} and B’={v1(-1, 2), v2(2, -2)} be bases for R2, and let −2 A= − be the matrix for T:R2→R2 relative to B find A’? Exp2) T: R2 → R3 is defined by T(x,y) = (x – y, 2x + 3y, -x) a) Find a basis for ker(T) and nullity(T) b) Find a basis for range(T) and rank(T) c) Find the standard matrix for T d) Find the matrix for T relative to B = {u1(1, -1), u2(0,1)} ⊂ R2 and B’ ={v1(1, 1, 1), v2(1, 1, 0), v3(1, 0, 0)} ⊂ R3 Exercise 1) T: R3 → R3 is defined by T(x,y,z) = (x – y + 2z, 2x + y - z, x + 2y + z) and the basis B = {u1(1, 0, 1), u2(0, 2, 2), u3(1, 2, 0)} Find the matrix for T relative to the basis B? Exercise 2) Let B = {u1(1, 0, 1), u2(0, 2, 2), u3(1, 2, 0)} and B’ = {v1(1, 1, -1), v2(1, 0, 1), v3(0, 2, 0)} be bases for R3, and let −1 A = −2 be the matrix for T:R3 → R3 relative to B find the matrix for T relative to B’ Exercise3) Given the two sets S = {u1(1,1,1), u2(1,1,0), u3(1,0,0)} and S’ = {v1(-6,-6,0), v2(-2,-6,4), v3(-2,-3,7)} in R3 1) Verify that the set S and S’ are bases for R3 2) Find the transition matrix from S to S’ 3) Find the transition matrix from S’ to S 4) Applying Gram – Schmidt process to transform the basis S into an orthonormal in R3 Exercise4: find a basis for and the dimension of the subspace of R3 spanned by S S = {u1(1,2,2), u2(-1,0,0), u3(1,1,1)} Exercise5: find an orthonormal basis for and the dimension of the solution space of the following system − x + y + z = =0 3 x − y 2 x − y − z = 4 x − y + z = 2 x + y − z = 3 x + y + z = CHAPTER IV: Eigenvalues and Eigenvectors 4.1 Eigenvalues and Eigenvectors 4.2 Diagonalization Keywords: eigenvalue, eigenvector, eigenspace, characteristic equation, characteristic polynomial, multiplicity, diagonalizable, diagonalization 4.1 Eigenvalues and Eigenvectors Definitions of Eigenvalue and Eigenvector Let A be an nxn matrix The scalar λ is called an eigenvalue of A if there is a nonzero vector x such that Ax = λx The vector x is called an eigenvector of A corresponding to λ Example 1: for the matrix 2 A= − verify that x1=(1, 0) is an eigenvector of A corresponding to the eigenvalue λ1 = 2, and that x2=(0,1) is an eigenvector λ2 = -1 Finding Eigenvalues and Eigenvectors Theorem: Let A be an nxn matrix 1) An eigenvalue of A is a scalar λ such that det(λI – A) = 2) The eigenvectors of A corresponding to λ are the nonzero solutions of (λI – A)x = In other words, the set of all eigenvectors of a given eigenvalue λ, together with the zero vector, is a subspace of Rn This subspace is called the eigenspace of λ Example 1: find the eigenvalues and corresponding eigenvectors of −12 A= − 2 0 A = 0 Note: If A is an nxn triangular matrix, then its eigenvalues are the entries on its main diagonal 2 0 A = −1 5 −3 −1 0 A= 0 0 0 0 0 0 −4 Example 2: find a basis for each of the corresponding eigenspaces −2 A = −2 0 1 A = 0 1 0 3 4.2 Diagonalization Problem: For a square matrix A, does there exist an invertible matrix P such that P-1AP is diagonal? You can review that two square matrices are similar if there exists an invertible matrix P such that B = P-1AP Matrices that are similar to diagonal matrices are called diagonalizable Definition of a diagonalizable matrix An nxn matrix A is diagonalizable if A is similar to a diagonal matrix That is, A is diagonalizable if there exists an invertible matrix P such that P-1AP is a diagonal matrix Example: verify that A is diagonalizable by computing P-1AP? 3 3 1 1) A = , P= −1 5 1 1 2 1 1 2) A = −1 , P = 0 −1 0 0 Have some problems Which square matrices are diagonalizable? Theorem: An nxn matrix A is diagonalizable if and only if it has n linearly independent eigenvectors p1, p2, …, pn How can you determine the matrix P? P = [p1 p2 … pn] Example Show that the matrix A is diagonalizable −1 −1 A = −3 −1 Find a matrix P such that P-1AP is diagonal? Theorem: If an nxn matrix A has n distinct eigenvalues, then the corresponding eigenvectors are linearly independent and A is diagonalizable Example: Determine whether the matrix A is diagonalizable 1 −2 A = 0 0 −3 Example: find an invertible matrix P such that P-1AP is diagonal 1 1) A = − −2 3) A = 0 1 0 2) A = 0 1 0 1 −1 4) A = −3 3 Exercise 1: a linear transformation T: R3 → R3 is given by T(x, y, z) = (x + 3y, y – 2z, -2x – 12z) 1) Find a basis for and the dimension of ker(T)? 2) Find the standard matrix for T? 3) Find a basis for and the dimension of range(T)? 4) Find the matrix A for T relative to B = {u1(1,1,1), u2(1,1,0), u3(1,0,0)} ⊂ R3 5) Use the matrix A to find T(1, -2, 0)? Exercise 2: a linear transformation T: R2 → R3 is given by T(x, y) = (x – y, 2x + 3y, -y) 1) Find the matrix A for T relative to B={u1(1,0), u2(1, 1)}⊂ R2 and B’={v1(1,0,1), v2(0,2,0), v3(-1,1,2)} ⊂ R3 2) Use the matrix A to find T(1, -2)? 3) Assume that a matrix Q is the matrix for T relative to two bases B ⊂ R2 and B’ ⊂ R3, where −1 Q = and B = {u1 (1, −1), u2 (1, 0)} ⊂ R −4 Find a basis B’ ⊂ R3 ... matrix, then the minor Mij of the element aij is the determinant of the matrix obtained by deleting the ith row and jth column of A The cofactor Cij is given by Cij = (-1)i+jMij Definitions of the... greater), then the determinant of A is the sum of the entries in the first row of A multiplied by their cofactors That is, n det( A) = ∑ a1 j C1 j = a11C11 + a12C12 + ⋯ + a1nC1n j =1 j= Theorem... 1) R2 with the standard operations is a vector space 2) Rn with the standard operations is a vector space 3) The set of all mxn matrices with the standard operations is a vector space 4) The set
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