2.8 Homework
MA 2321, Fall '03, T. Olson

7.

Let $\vec{v_1}= \left( \begin{array}{r} 2\\ -8\\ 6 \end{array} \right)$, $\vec{v_2}= \left( \begin{array}{r} -3\\ 8\\ -7 \end{array} \right)$, $\vec{v_3}= \left( \begin{array}{r} -4\\ 6\\ -7 \end{array} \right)$, $\vec{p}= \left( \begin{array}{r} 6\\ -10\\ 11 \end{array} \right)$, and $A = \left( \vec{v_1} \quad \vec{v_2} \quad \vec{v_3} \right)$.

a.

How many vectors are in the set $\{\vec{v_1},\vec{v_2},\vec{v_3}\}$?

b.

How many vectors are in Col A?

c.

Is $\vec{p}$ in Col A? Why or why not?

11.

Give integers p and q such that Nul A is a subspace of $\mbox{I\hspace{-.14em}R}^p$ and Col A is a subspace of $\mbox{I\hspace{-.14em}R}^q$ for the matrix

$$A = \left( \begin{array}{rrrr} 3 & 2 & 1 & -5 \\ -9 & -4 & 1 & 7 \\ 9 & 2 & -5 & 1 \end{array} \right) .$$

13.

For A as in Exercise 11, find a nonzero vector in Nul A and a nonzero vector in Col A.

23.

The following displays a matrix A and an echelon form of A.

$$A = \left( \begin{array}{rrrr} 4 & 5 & 9 & -2 \\ 6 & 5 &... ... 0 & 1 & 5 & -6 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)$$

Find a basis for Col A and a basis for Nul A.

27.

Construct a $3\times 3$ matrix A and a nonzero vector $\vec{b}$ such that $\vec{b}$ is in Col A, but $\vec{b}$ is not the same as any one of the columns of A.

28.

Construct a $3\times 3$ matrix A and a vector $\vec{b}$ such that $\vec{b}$ is not in Col A.

29.

Construct a nonzero $3\times 3$ matrix A and a nonzero vector $\vec{b}$ such that $\vec{b}$ is in Nul A.

33.

If Q is a $4\times 4$ matrix and Col Q $=\mbox{I\hspace{-.14em}R}^4$, what can you say about solutions of equations of the form $Q\vec{x}=\vec{b}$ for $\vec{b}$ in $\mbox{I\hspace{-.14em}R}^4$?

35.

What can you say about Nul B when B is a $5\times 4$ matrix with linearly independent columns?