# Practice Exam 2 M314

Note: These are mostly problems I've used on old exams. Thus it is longer than your actual exam will be, but this is a good thing since it gives you practice on more problems. Be sure also to study the problems on the syllabus, the quizzes, and what was covered in class. To save space, column vectors are sometimes given as the transpose of a row vector.

[1] (6 points) Let A be an n x n matrix. If the equation Ax = 0 has only the trivial solution, do the columns of A span Rn? Why or why not?

[2] (8 points) Let B and C be 4 x 4 matrices such that det B = 2 and det C = -5, and let A be the matrix
```[ 0   4   5   1 ]             (a) Determine det (3(B-3C2)T).
[ 1   0   7   1 ]
[ 1   0   2   1 ]
[ 0   1   3   0 ].            (b) Use Cramer's rule to find x2 in the equation
Ax = b, where bT = ( 5, 7, 2, 3). (This can be done
(c) Why can't you always use Cramer's rule to solve linear equations? Give an example
of a linear system of equations for which you can't use Cramer's rule, and explain
why Cramer's rule doesn't work for your example.
```
[3] (6 points) The nullspace of an m x n matrix A (with real number entries) is a subset of Rn; prove that it is a subspace. (Mention the three things you have to check, and then show how to check them.)

[4] (9 points) For this problem, you may assume that B is the reduced row echelon form of A, where A and B are the following matrices:
```[ 3  1 -2  0  1  2  1 ]        [ 1  0  -1   0   0  -2  -3 ]   (a) Find a basis for the
[ 1  1  0 -1  1  2  2 ]        [ 0  1   1   0   0   2   3 ]       column space of A.
[ 3  2 -1  1  1  8  9 ]        [ 0  0   0   1   0   4   5 ]
[ 0  2  2 -1  1  6  8 ]        [ 0  0   0   0   1   6   7 ]   (b) Find a basis for the
[ 0  3  3  3 -3  0  3 ]        [ 0  0   0   0   0   0   0 ]       null space of A.
```
[5] (6 points) Find the coordinates of ( 1, 2)T with respect to the basis {( 1, 4)T, ( 1, 1)T} of R2.

[6] (6 points): Say A is a 5x7 matrix with dim Nul(A)=4. Find the dimension of the row space of A and of the null space of AT.

[7] (6 points): Let A be the matrix
```                  [ 1  -4   3 ]
[ 4  -4   0 ]
[ 2  -4   2 ].
```
In each case determine if the given vector is an eigenvector of A and if so find its eigenvalue.
• (a) v = [1,1,1]T
• (b) v = [1,0,1]T
• (c) v = [0,0,0]T
[8] (10 points): Let A be the matrix
```                  [ 1  1  1 ]
[ 0  2  1 ]
[ 0  0  3 ].
```
This is a diagonalizable matrix; thus P-1AP=D for some invertible matrix P, where D is a diagonal matrix.
• (a) Find each eigenvalue l of A, and a basis for each eigenspace El.
• (b) Find P and D.
[9] (8 points): For each of the matrices below, determine if it is diagonalizable. Give a reason why or why not. (These can all be done by inspection.)

```(a) The 5x5 identity matrix.          (b) [ 1  1  0  0  0 ]
[ 0  1  1  0  0 ]
[ 0  0  1  0  0 ]
[ 0  0  0  0  1 ]
[ 0  0  0  0  0 ]

(c) A 5x5 matrix A with eigenvalues    (d) A matrix of the form PDP-1, where
0, 2, and 3, and such that the             P is invertible and D is diagonal.
dimension of the null space is 2
for both A and A-3I.
```
[10] (8 points): Let A be the matrix
```                  [ 0  -1 ]
[ 2   3 ].
```
• (a) Find the eigenvalues of A.
• (b) Find a basis for the eigenspace corresponding to each eigenvalue of A.
• (c) Find A10v, where v = [1, 0]T.