sample linear algebra questions

8/13/2019 Sample Linear Algebra Questions

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Sample Linear Algebra Questions

1. Solve for x1, x2 andx3 in the following matrix equation.

x1 x2 x3

2 1 1

1 3 1

1 2 1

= 5 1 1

2. Express the vector u =

1

2

3

as a sum of vectors v and w, where v

belongs to the subspace S=

x

y

z

x+y+z = 0

andw belongs toS, the

subspace orthogonal to S.

3. What are the eigenvalues of a 2-by-2 matrix whose trace is 1 and determi-

nant is -12?

4. Suppose matrixA is reduced by the usual row operations to the matrix

R=

1 4 0 2

0 0 1 2

0 0 0 0

Find all solutions x R4 (if the solution exists) to the equation

Ax= sum of the columns ofA

(Strang)

5. Let A be a matrix such that

A

1

2

2

=

3

6

1


2/4

and

A

3

2

1

= 44

.

Find

A

5

6

3

6. Find the least squares line for the three points (1, 1), (1, 5), (3,3).

7. Let A be a 3-by-3 anti-symmetric matrix, that is, AT =A. What can be

said about the determinant ofA? What can be said about the determinant of

an n by n antisymmetric matrix where n is an odd positive integer?

8. Given that a four by four matrix A is reduced to the upper triangular

matrix

1 1 3 3

0 1 2 1

0 0 1 2

0 0 0 1

by the following sequence of row operations, find matrix A.

1) 2R1+R2 R2

2) 4R1+R3 R3

3) 7R1+R4 R4

4) 13R2 R2

5) 5R2+R3 R3

6) 8R2

+R4

R4

7) 16R3 R3

8) 111R4 R4

2


3/4

9. Consider the system

ax + y = a2

x + ay = 1

. For what values ofa are there infinitely many solutions, no solutions, and a

unique solution?

(From USSR Olympiad)

10. Categorize the following subsets ofR3 as subspaces or not subspaces.

a. {(a,b, 3a2b)}

b. {(a,b,c) | a+b+c= 0}

c. {(a,b,c) | ab= bc}

d. {(a,ab,b)}

e. {(a,b,a+b+ 1)}

f. {(a+ 2, b, 2ab+ 4)}

11. Solve forx2 using Cramers rule.

2x1 + x3 + x4 = 5

x2 x4 = 13x1 x3 x4 = 0

4x1 + x2 + 2x3 + 3x4 = 9

12. Define the matrix

An=

1

6 0 1

6

0 1 01

6 0 1

6

n

Find and prove limnAn.

13. Let P2 be the vector space of all polynomials of order two or less over the

real numbers. That is , P2 = {ax2 +bx+c | a,b, c R}. Consider a function

T : P2 P2 defined as

T(p(x)) = xp(x)

3


4/4

for any p(x) P2. Prove that T is a linear function (linear transformation).

Find the kernel of T. Also, find the eigenvalues and eigenvectors ofT.

14. Find bases for the row space, column space, and null space of the following

matrixA.

A=

1 1 2 3

2 1 5 2

3 1 8 1

. Confirm the rank nullity theorem using matrix A.

15. Suppose u, v, and w are three vectors from R3 such that

det

u

v

w

= 10

Find the following

det

v

u

w

det

2u

vw

det

u

v

2u + 3v

4

sample linear algebra questions

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