LINEAR ALGEBRA [07EC10] DEPARTMENT OF ELECTRONICS AND COMMUNICATIONS MODEL QUESTIONS

UNIT-1 1.Write down a 2 X 2 system with (a) A unique solution (b) No solution (c) Infinitely many solutions 2.Given the matrices A = 1

0

0

1

1

0

1

and

B = 1

1

0

0

0

1

1

1

0

0

1

Compute AB, BA, A-1, B-1 and (AB)-1 . What do you infer? 3.Relate the rows of EA with the rows of A, where E

= 1

0

0

0

2

0

4

0

1

4. Find the inverse of A( if it exists) by Gauss Jordan method. 5.Factorize the matrix A = 1

0

1

1

1

0

0

1

1

into LU or PA = LU.

6.For which value of k does the system kx + y = 1 x + ky = 1 has (a)no solution (b)a unique solution

(c)infinitely many solutions?

1

7. Solve Ax = b by solving the triangular systems Lc = b and Ux = c A =

1

0

O

2

2

4

b =

0

4

1

0

0

1

3

0

1

0

1

0

0

1

1

8.Find the symmetric factorization A = LDLT of the matrix A =

1

2

0

2

6

4

0

4

11

9. Solve by elimination of the system: u+v+w=0 u + 2v + 3w = 0 3u + 5v + 7w = 1 10. True or false, with reason if true and counter example if false: (a) If L1U1 = L2U2, then L1 = L2 and U1 = U2 ie., LU factorization is unique. (b) If A2 + A = I then A-1 = A + I (c) If all diagonal entries of A are zero, then A is singular. (d) If A and B are invertible then BA is invertible. 11. Find

2

3

2

3

0

0

0

1

12. Find all matrices A = a

b

that satisfy

c

d

A 1

1

= 1

1

1

1

n

,

n

,

2

3

0

1

-1

1 A 1

13. Describe the row and column pictures for 2x – y = 1 , x + 2y = 6. What is the solution?

2

14. Suppose we solve Ax = b for three special RHS b: Ax1 = 1

and Ax2 = 0

Ax3 = 0

0

1

0

0

0

1

If the solutions x1,x2, x3 are the columns of a matrix X. Find AX. 15. Solve for the columns of A-1 = x

t

y

z

and

10

20

t

= 0

20

50

z

1

10

20

x

= 1

20

50

y

0

16.Given that A = a

b

b

a

a

b

a

a

a

show that A is invertible whenever a≠ 0 and a ≠ b. Find A-1. 17.Show that A2 = 0 is possible, but ATA is impossible unless A = 0. 18.Prove that A = 1

1

3

3

is not invertible by failing to solve

1

1

a

b = 1

0

3

3

c

d

1

0

19.Give examples of A and B such that (a) A + B is invertible although A and B are not invertible. (b) A + B is not invertible, but A and B are invertible. (c) A, B and A + B are invertible. 20. Given A = 1 3

and B = 3 , compute ATB, BTA, ABT and BAT. 1

3

21. If A is m x n, B is p x n, C is n x p, D is m x p, then which of these matric multiplications are allowed and what are their orders? AB, BA, ABC, BAC, CD, DB, CB, AC. 22. Which 3 matrices E21, E31, E32 put A into upper triangular form U? A = 1

4

-2

1

6

2

0

1

0

Also find U.

23. Compute the products: 2

1

1

0

3

1

and

2

0

1

1

3

1

Represent it pictorially. What is your conclusion? 24. Working a column at a time, compute the products 4

1

1

5

1

3

6

1

and

4

5

6

1

1

1

1

2 3

25. The pivots (1,4), (2,8) and (3,14) lies on the parabola y = a + bx + cx2. Find the constants a, b and c. 26. If the age of X is twice that of Y and their ages add up to 39, then find their ages graphically. 27. If the 3 columns of B are the same, then show that the 3 columns of EB are the same. Why? 28. True or False: (a) If A2 is defined then A has to be a square matrix. (b) If AB = A, then B = I. (C) If AB and BA are defined then AB and BA are square matrices. (d) If AB and BA are defined then A and B are square matrices.

4

29. If P is a permutation matrix of order 5 x 5, why is P6 = I? Also find a non-zero vector x so that (I – P)x = 0. 30. Solve using Gauss-Jordan method: 1

a

b

0

1

c

0

0

1

x1

x2

1

0

0

x3 = 0

1

0

0

0

1

UNIT : 2 1.Do the vectors (1, 1, 3), (2, 3, 6) and (1, 4, 3) form a basis for R3? 2. In R2 define addition and scalar multiplication as follows: (x1, y1), (x2, y2) Є R2 (x1, y1) + (x2, y2) = (x1 + x2 + 1, y1 + y2 + 1) α (x1, y1) = (αx1, αy1) Which of the 8 rules are broken? 3. Which of the following are subspaces of R∞? (a) All increasing sequences: xj + 1 ≥ xj for each j. (b) all vectors (b1, b2,…..) such that

𝒂 𝒊=𝟏 𝒃i

=0

(c) all vectors (b1, b2,……) such that b1.b2…… = 0 4. Let A be a 10 x 10 invertible matrix. What is C(A)? What is N(A)? Why? 5.Describe the column spaces of the matrices: A = 1 2

0 0

0

B = 1

0

0

C = 1

2

0

0

0

2

0

0

0

0

6. True or False: (with counter example if false) (a) The vectors b that are not in the column space C(A) form a subspace. (b) Column space of 3A equals column space of 4A. (C) If A is a zero matrix the C(A) = { 0 }

5

(d) C(A) = C(B)  A = B 7. Construct a subset of R2 that is: (a) closed under scalar multiplication but not under vector addition. (b) closed under vector addition but not closed under scalar multiplication. (c) not closed under vector addition and scalar multiplication. 8. Find the column space and null space of: A = 1

2

1

2

6

3

0

2

5

9. Under what conditions on the b vector is the system solvable? Find the solution. 1

2

x1 =

b1

2

4

x2

b2

2

5

b3

3

9

b4

10. Choose three independent columns of U and A: A = 2

3

4

1

4

6

8

0

0

0

6

and

U = 2

3

4

1

2

0

6

7

0

0

9

0

0

0

9

7

0

0

0

0

0

11. By solving c1v1 + c2v2 + c3v3 + c4v4 = 0, show that 1 0 0

1 ,

1 0

,

1

1

1 ,

4

1

2

are linearly independent.

6

12. Let w1, w2, w3 be 3 independent vectors. Define new vectors v1 = w1, v2 = w1 + w2, v3 = w1 + w2 + w3. What can you say about the vectors v1, v2, v3? If dependent then find a combination of v1, v2, v3 that give zero. 13. Let w1, w2, w3, w4 be 4 independent vectors . Define v1 = w1 – w2, v2 = w1 – w3, v3 = w1 – w4, v4 = w2 – w3, v5 = w2 – w4, v6 = w3 – w4. Are these new of vectors independent or dependent? In case they are dependent find a combination of vectors that give zero. 14. Given that A =

1

0

1

3

1

3

2

1

2

and

U = 1

0

1

0

1

0

0

0

0

Find bases for the two column spaces and the two null spaces. 15. True or False: (give reasons) (a) If the columns of a matrix are dependent so are the (b)The columns of a matrix A are a basis for C(A). (c)The null space of the identity matrix I3 is the whole of R3. (d) column space of a 3 x 3 diagonal matrix is whole of R3 (e)If A is 2 x 2 matrix then column space of A is equal to its row space. 16. Describe the f fundamental sub-spaces associated with

1

0

0

0

1

0

0

0

0

17. Find the dimension and a basis for the four fundamental subspaces for A = 1

0

1

2

1

0 1

and

U = 1

0

1

2

0

1

0

1

0

0

0

0

0

1

0

0

0

18. Construct a basis for the four sub-spaces associated with the matrices:

7

A = 0

0

1

and

U = 0

0

2

1

2

4

8

0

0

0

0

0

0

19.Find the rank of A and write A = uvT: A = 1

0

2

0

0

0

0

0

0

3

0

6

20.Without actually computing A, find bases for the four fundamental subspaces: A = 1

0

0

1

2

3

4

6

1

0

0

1

2

3

9

8

1

0

0

1

2

B = 1

3

2

5

4

12

21.If AB = 0 then show that r(A) + r(B) ≤ n 22. Find the left inverse/ right inverse of: A =

1

2

3

2

3

4

and

23.The solutions to the linear differential equation …. form a vector space. Find a basis for the solution space. 24. From the cubics P3 to the fourth degree polynomials P4. What matrix represents multiplication by (3 + 5t)? 25. Which of the following transformations are not linear? Assume v = (v1, v2) (a)T(v) = (v2, v1)

(b)T(v) = (1, 0)

8

(c) T(v) = (v2, v2)

(d)T(v) = (v1, 0)

26.Find the 4 x 3 matrix for the right shift transformation: (x1, x2, x3)  (0, x1, x2, x3) Also find the left-shift matrix from R4 to R3 that is obtained from the transformation: (x1, x2, x3, x4)  ( x2, x3, x4). Find AB and BA. 27. Find T(T(v)) for the following transformations: (a) T(v) = -v (b) T(v) = v + (1,1) (c) T(v) = (v1, 0) (d) T(v) = (v2, v1) 28. Find the range and kernel of T: (a) T(v1, v2) = (v2, v1) (b) T(v1, v2) = (v2, v2) (c) T(v1, v2, v3) = (v1, v2) (d) T(v1, v2) = (0,0) 29. What matrix transforms (a) (1,0) and (0,1) to (1,4) and (1,5) (b)(1,4) and (1,5) to (2,3) and (3,4) (c)(1,0) and (0,1) to (2,3) and (3,4) 30. There are 2 bases : v1, v2 ,…..vn and w1, w2,….wn for Rn. If a vector x Є Rn is such that x = b1v1 + b2v2 + …+ bnvn and x = c1w1 + c2w2 +….. +cnwn then what is the change of basis matrix? 31. What is the effect of 10 reflections and 8 rotations of the x-y plane? Is it a reflection or a rotation? 32. The matrix A = 3

0

0

1

produces a stretching in the x – direction.

9

What is its effect on any vector (x,y) Є R2? With x2 + y2 = 2, What is the shape of the new curve? 33. Let A = 1

2

3

6

then show that the identity matrix.

I is not in the range of T. Find M ≠ 0 such that T(M) = AM is zero. 34. Which of the transformations satisfy T( u + v) = T(u) + T(v) and which satisfy T(cu) = cT(u)? (a) T(u) = u1 + u2 + u3 (b) T(u) = (u1, 2u2, 3u3) (c) T(u) = min ui

(1 ≤ i ≤ 3)

(d) T(u) = (u1 + 1, u2 – 1) 35. T transforms (1,1) to (2,2) and (2,0) to (0,0). Find T( ) When (a) v = (2,2), (b) v = (3,1) , (c) v = (-1, 1), (d) v = (a, b) 36. If S =

cos2θ

sin2θ

and

Sin2θ

-cos2θ

T = cos2α sin2 α

sin2 α -cos2 α

Are 2 reflections then what is the product ST? 37. The matrix M = r

s

t

u

respectively. The matrix N =

transforms (1,0) to (r,t) and (0,1) to (s,u)

a

b

c

d

-1

transforms (a, c) to (1,0) and (b, d) to (0,1). How do you transform (a, c) to (r, t) and (b, d) to (s, u)? what is the matrix of this transformation? 38. Let {[1 0 0]T, [0 1 0]T, [ 0 0 1]T} and {[-1 2 2]T, [2, 2 -1]T, [ 2 -1 2]T} be two bases for for R3. Compute the metrix of transformation from one basis to the other.

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