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1.  If \(A\) is a 3 by 3 matrix such that the null space is a line through the origin, then the column space is

a.  just the zero vector

b.  a line through the origin

c.  a plane through the origin

d.  all of \({\bf R}^3\)

e.  insufficient information to say

2.  If \(A\) is a 3 by 3 matrix such that the null space is a line through the origin, then the determinant is

a.  0

b.  1

c.  2

d.  3

e.  insufficient information to say

3.  If \({\bf v}, {\bf w}\) are linearly independent vectors in \({\bf R}^3\), then \(\{c_1{\bf v} + c_2{\bf w}\), where \(c_1\) and \(c_2\) are real numbers\(\}\) constitutes a subspace of \({\bf R}^3\).

a.  Definitely false

b.  Definitely true

c.  There is not enough information to say.

4.  If \({\bf v}, {\bf w}\) are linearly indpendent vectors in \({\bf R}^3\), then \(\{c_1{\bf v} + {\bf w}\), where \(c_1\) is a real number\(\}\) constitutes a subspace of \({\bf R}^3\).

a.  Definitely false

b.  Definitely true

c.  There is not enough information to say.

5.  Is \(\{\left[\begin{array}{c}x_1\\x_2\end{array}\right]: (x_1)(x_2) \geq 0\}\) a subspace of \({\bf R}^2\)?

a.  No because it doesn't contain the zero vector.

b.  No because it isn't closed under vector addition.

c.  No because it isn't closed under scalar multiplication.

d.  Yes it is.

6.  Say \(A\) is a 5 by 6 matrix.

a.  The null space is \({\bf R}^5\).

b.  The null space is just the zero vector.

c.  Neither of the above can be true.

d.  There is not enough information to say.

7.  Say \(A\) is a 5 by 6 matrix.

a.  The column space is \({\bf R}^5\).

b.  The column space just the zero vector.

c.  Neither of the above can be true.

d.  There is not enough information to say.