Linear algebra exercises

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Contents

1 MATRICES AND OPERATIONS ccc cccececcceeeeeeeeeevreereeveenes 3

Pa ĐI) 010) kaiiẳầađiaẳđađđ 9

3 THE INVERSE MATRIN 020cc n nnnn nh ng nh nh nh nà 14

4, RANK OF MATRIX .cccccccccccececeevtereevteureveterevreereerenees 20

5 SOLVE GENERAL LINEAR SYSTEM c 222cc sài 23

6 SOLVE LINEAR SYSTEM WITH PARAMETERS 27

7 LINEAR INDEPENDENCE cuc c0 6c n2 ng nh nh nh xu 31

8 SUBSPAGE, BASIS AND DIMENSION c2 cà 36

9 LINEAR BOUND AND SOLUTION SPACE c.ccccccccc: 38

10 MATRIX OF LINEAR MAP c2 2n ccnn nh nh nh he 41

11 EIGENVALUES AND EIGENVECTORS c2 cà 43

12 VECTOR COORDINATES c0 2n nnn nh nh kh xu 46 IEĐF.\€.9)V.00v420E.017.vi:iaađaađaaađ 49

1 MATRICES AND OPERATIONS

3 THE INVERSE MATRIX

C A has an inverse matrix for all m

D A has an inverse matrix when m = 2

5 SOLVE GENERAL LINEAR SYSTEM

A (a,b, —2a, —2b + 1, a) Va, b

7 Solve the given system

A The system does not have solution

8 Solve the given system

A The system does not have solution

x=1—-2a, y=2-3a, z=a, Va

x=-l-a, y=—-6+a, z=a, Va

œ=—l— 2a, y=-6- 3a, z=a, Va

12 Which system has trivial solution?

It has an unique solution

It does not have solution

It has only 2 solutions

The set of its solution is {(3a, —a, 2a), Va}

It has only trivial solution

The set of its solution is {(2a, —a, a), Va}

6 SOLVE LINEAR SYSTEM WITH PARAMETERS

tT +mxr = 0

+ 3nxr = 0

A It has non-trivial solution when m = 3n

B It has an unique solution when m = 3n

C It has infinite solutions when m F 3n

D It does not have solution when m> 0

3 Find m such that {

A It has an unique if and only if m # 1

B It does not have solution when m = —1

C It has solution if and only ifm A +1

D It has solution for all m

7 LINEAR INDEPENDENCE

1 In R®, let v = (2,m,1); ơị = (0,2,3); vg = (1,5,2).Find m such that v is a

linear combination of ¡ and 2

D There doesn’t exist m

3 In R®, let v, = (—2,1,3); ve = (1, —4,6); v3 = (2m,2,m+10).Find m such that

U1, v2, U3 is linear dependence

4, In R?, which system is linear dependence?

A System A and B are linear independence

B A is linear independence, P is linear dependence

C B is linear independence, A is linear dependence

D System A and B are linear dependence

6 Find m such that {u = (1,1,1),v = (m,1,1),w = (2,m,—1)} is linear indepen- dence

8 In Rỷ, let z = (1,3,5); = (3,2,5); v = (2,4,7) and w = (5,6,k).Find k such

that x is a linear combination of u,v and w

A x and y are linear combinations of {u,v}

B x is a linear combination of {u,v}, y is not a linear combination of {u,v}

C x and y are not linear combinations of {u,v}

D x is not a linear combination of {u,v}, y is a linear combination of {u,v}

10 In R°, rank of the system {(1, —2, 3); (—2,3,4);(—1,1,7)} is

11 In R’, let {v, = (1,-2,3); ve = (—2,3,4); v3 = (—1,1,m).Find m such that

rank of the given system is equal to 2

15 Find the condition of (x1, 22,73) such that it is a lmear combination of

17 Let wu, u2,u3 is linear independence in R* Which statement is TRUE?

{u1, U2, O} is linear dependence

{u1, 2, u3,0} is linear dependence

18 Find the rank of the given system below

{ii = (3,1,5,7), ue = (4,-1,-2,2), ug = (10,1,8,17), us = (13, 2,13, 24)} A.r=1

8 SUBSPACE, BASIS AND DIMENSION

D There does not exist any value of m

9 LINEAR BOUND AND SOLUTION SPACE

1 In R®, let the subspace W generated by the vector system {(1,—2,3), (—2,4—

dim W is the smallest

4 In RÝ, let the subspace L, generated by the vector system {(1,2, —1,0), (1, —1,2, 1)},

find m such that (2,m,1,m)} belongs to the subspace L

10 MATRIX OF LINEAR MAP

11 EIGENVALUES AND EIGENVECTORS

Find all eigenvectors corresponding to the eigenvalue \ = 1 of A =

13 DIAGONALIZE A MATRIX

0 -—m m0

A A is diagonalizable if and only if m = 0

B A is not diagonalizable if and only if m = 0

C A is diagonalizable for all m

D A does not have any eigenvalue

C A is diagonalizable and B is not diagonalizable

D A and B are not diagonalizable

A M is diagonalizable if and only if m = 0

B M is not diagonalizable if and only if m = 0

C M is diagonalizable for all m

D M has only one eigenvalue

A M is diagonalizable if and only if a = b= 0

B M is not diagonalizable if and only if a = 0

C M is diagonalizable where a, b are arbitrary

D M is not diagonalizable for all a, }

A M is diagonalizable if and only if m = 0

B M is not diagonalizable if and only if m = 1

C M is diagonalizable where m is arbitrary

D M is not diagonalizable for all m

A M is diagonalizable if and only if m = 0

B M is not diagonalizable if and only if m = 0

C M is diagonalizable where m is arbitrary

D M is not diagonalizable for all m