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For questions 1 through 5, let $$A = \left[\begin{array}{ccccc} 1 & 2 & 1 & 1 & -2\\ 2 & 4 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & -2 & 5\end{array}\right]$$

1.  Find a basis for Nul(\(A\)).

a.  $$\left\{ \left[ \begin{array}{c} -2\\1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\-1\\2\\1 \end{array}\right]\right\}$$

b.  $$\left\{ \left[ \begin{array}{c} -2\\1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2\\-1 \end{array}\right]\right\}$$

c.  $$\left\{ \left[ \begin{array}{c} -2\\1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2\\-1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\1\\0\\-1 \end{array}\right]\right\}$$

d.  $$\left\{ \left[ \begin{array}{c} -2\\1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2\\-1 \end{array}\right], \left[ \begin{array}{c} 0\\3\\1\\0\\-2 \end{array}\right]\right\}$$

2.  Find a basis for Col(\(A\))

a.  $$\left\{ \left[ \begin{array}{c} 1\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 0\\1\\0\\0 \end{array}\right], \left[ \begin{array}{c} 0\\0\\1\\0 \end{array}\right]\right\}$$

b.  $$\left\{ \left[ \begin{array}{c} 2\\0\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\-1\\2\\0 \end{array}\right]\right\}$$

c.  $$\left\{ \left[ \begin{array}{c} 2\\4\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2 \end{array}\right]\right\}$$

d.  $$\left\{ \left[ \begin{array}{c} 1\\2\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\1\\0\\1 \end{array}\right], \left[ \begin{array}{c} 1\\0\\1\\-2 \end{array}\right]\right\}$$

e.  $$\left\{ \left[ \begin{array}{c} 1\\2\\0\\0 \end{array}\right], \left[ \begin{array}{c} 2\\4\\0\\0 \end{array}\right], \left[ \begin{array}{c} 1\\1\\0\\1 \end{array}\right], \left[ \begin{array}{c} -2\\-1\\-2\\5 \end{array}\right]\right\}$$

3.  Find a basis for Row(\(A\)). (Take the non-zero rows of your reduced-echelon form matrix)

a.  $$\left\{ \left[ \begin{array}{c} 1\\2\\1\\1\\-2 \end{array}\right], \left[ \begin{array}{c} 2\\4\\1\\0\\-1 \end{array}\right]\right\}$$

b.  $$\left\{ \left[ \begin{array}{c} 1\\2\\0\\0\\-1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\1\\0\\1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\0\\1\\-2 \end{array}\right]\right\}$$

c.  $$\left\{ \left[ \begin{array}{c} 1\\2\\0\\0\\-1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\1\\0\\1 \end{array}\right], \left[ \begin{array}{c} 0\\0\\0\\1\\2 \end{array}\right]\right\}$$

4.  What is Rank(\(A\))? (This is just the number of vectors in a basis for the column space.)

a.  2

b.  3

c.  4

5.  Is \(A\) invertible?

a.  Yes.

b.  Go jump in the lake.

6.  Say \(A\) is a 5 by 8 matrix. If the dimension of Row(A) is 2, what is the dimension of Col(A)?

a.  2

b.  3

c.  5

d.  6

7.  Say \(A\) is a 5 by 8 matrix. If the dimension of Row(A) is 2, what is the dimension of Nul(A)?

a.  2

b.  3

c.  5

d.  6

8.  Find all the eigenvalues of the matrix $$A = \left[\begin{array}{cc} 1 & 2\\ 2 & 4\end{array}\right]$$

a.  0 and 5

b.  0 and 1 and 2

c.  1 and 5

d.  1 and 2 and 4

For questions 9 through 11, let $$A = \left[\begin{array}{cccc} 1 & 2 & 3 & 4\\ 0 & 1 & 2 & 0\\ 0 & 0 & 2 & 1\\ 0 & 0 & 0 & -2\end{array}\right]$$

9.  Find the eigenvalues for \(A\).

a.  1, 2, 3, 4

b.  0, 1, 2, 3, 4

c.  -2, 0, 1, 2, 3, 4

d.  -2, 1, 2

e.  0, 1, 2

10.  Describe the eigenspace for \(\lambda = 1\).

a.  A line through the origin

b.  Two lines through the origin

c.  A plane through the origin

d.  All of \(R^4\)

e.  Just the zero vector

11.  Which of the standard unit basis vectors are eigenvectors for \(A\).?

a.  Just \(e_1\)

b.  Just \(e_4\)

c.  All of them.

d.  None of them.

12.  Let \(\left[\begin{array}{cc} 1 & 1 \\ 0 & 1\end{array}\right]\). Is \(A\) diagonalizable?

a.  Yes.

b.  No, because it has too few eigenvalues.

c.  No, because there aren't two linearly independent eigenvectors.

13.  Let \(\left[\begin{array}{cc} 1 & 0 \\ 0 & 1\end{array}\right]\). Is \(A\) diagonalizable?

a.  Yes.

b.  No, because it has too few eigenvalues.

c.  No, because there aren't two linearly independent eigenvectors.

14.  Say \(T:{\bf R}^2 \rightarrow {\bf R}^2\) is the linear transformation which reflects the plane over some line through the origin (you don't need to know which line, and you don't need to know the 2 by 2 matrix \(A\) such that \(T(\vec{x}) = A\vec{x}\)). What are the eigenvalues of T (Just think about it)?

a.  There are none

b.  Only 1

c.  Only -1

d.  1 and -1

c.  1 and 0

15.  Say \(T:{\bf R}^2 \rightarrow {\bf R}^2\) is the linear transformation which rotates the plane by 45 degrees around the origin. What are the eigenvalues of T (Just think about it)?

a.  There are none

b.  Only 1

c.  Only -1

d.  1 and -1

c.  1 and 0

16.  Say \(T:{\bf R}^2 \rightarrow {\bf R}^2\) is the linear transformation which rotates the plane by 180 degrees around the origin. What are the eigenvalues of T (Just think about it)?

a.  There are none

b.  Only 1

c.  Only -1

d.  0 and 1

c.  1 and 0

17.  Say \(T:{\bf R}^3 \rightarrow {\bf R}^3\) is the linear transformation which rotates three dimensional space by 45 degrees around some line through the origin. What are the eigenvalues of T (Just think about it)?

a.  There are none

b.  Only 1

c.  Only -1

d.  1 and -1

c.  1 and 0

18.  Consider the triangle with vertices A(1,2,3), B(5,5,3), and C(3,4,4). Find cos\(\theta\) where \(\theta\) is the smallest angle in the triangle.

a.  -14/15

b.  -1/15

c.  0

d.  1/15

e.  14/15