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

2.

If defined, compute the product shown using (i) the definition (linear combinations of columns) and (ii) the rule for computing $A\vec{x}$. If the product is undefined, say why.

$$\left( \begin{array}{rr} 4 & 1 \\ 2 & -5 \\ 0 & 6 \\ \e... ...} \right) \left( \begin{array}{r} 5 \\ 2 \end{array} \right)$$

4.

If defined, compute the product shown using (i) the definition (linear combinations of columns) and (ii) the rule for computing $A\vec{x}$. If the product is undefined, say why.

$$\left( \begin{array}{rr} 3 & -8 \\ -1 & 5 \\ 2 & -3 \\ ... ...ht) \left( \begin{array}{r} -4 \\ 1 \\ 0 \end{array} \right)$$

8.

Write the system in the form $A \vec{x} = \vec{b}$.

$$\begin{eqnarray*}3x_2 - 2 x_3 + x_4 & = & 0 \\ x_1 - 2x_2 + 6 x_3 & = & 0 \\ 7 x_1 + x_2 - 5x_3 - 8x_4 & = & 0 \end{eqnarray*}$$

19.

Let $$A= \left( \begin{array}{rr} -3 & 1 \\ 6 & -2 \end{array} \right)$$ and $$\vec{b}= \left( \begin{array}{r} b_1 \\ b_2 \end{array} \right)$$. Show that the equation $A \vec{x} = \vec{b}$ is not consistent for all possible $\vec{b}$, and describe the set of all $\vec{b}$ for which $A \vec{x} = \vec{b}$ is consistent.

31.

It can be shown that $$\left( \begin{array}{rrr} 4 & 1 & 2 \\ -2 & 0 & 8 \\ 3 & 5 &... ...\end{array} \right) = \left( \begin{array}{r} 4 \\ 18 \\ 5 \end{array} \right)$$. Use this fact (and no row operations) to find scalars c₁, c₂, and c₃, such that

$$\left( \begin{array}{r} 4 \\ 18 \\ 5 \end{array} \right) = c... ..._3 \left( \begin{array}{r} 2 \\ 8 \\ -6 \end{array} \right) .$$

32.

Let $\vec{u} = \left( \begin{array}{r} 3 \\ 8 \\ 4 \end{array} \right)$, $\vec{v} = \left( \begin{array}{r} 1 \\ 3 \\ 1 \end{array} \right)$, and $\vec{w} = \left( \begin{array}{r} 1 \\ 1 \\ 3 \end{array} \right)$. It can be shown that $2\vec{u}-5\vec{v}-\vec{w}=0$. Use this fact (and no row operations) to solve the equation $$\left( \begin{array}{rr} 3 & 1 \\ 8 & 3 \\ 4 & 1 \\ \end{a... ...end{array} \right) = \left( \begin{array}{r} 1 \\ 1 \\ 3 \end{array} \right)$$.

35.

Let A be a $3 \times 4$ matrix, let $\vec{y_1}$ and $\vec{y_2}$ be vectors in $\mbox{I\hspace{-.14em}R}^3$, and let $\vec{w}=\vec{y_1}+\vec{y_2}$. Suppose that $\vec{y_1}=A \vec{x_1}$ and $\vec{y_2}=A \vec{x_2}$ for some vectors $\vec{x_1}$ and $\vec{x_2}$ in $\mbox{I\hspace{-.14em}R}^4$. What fact allows you to conclude that the system $A \vec{x} = \vec{w}$ is consistent? (Note, $\vec{x_1}$ and $\vec{x_2}$ denote vectors, not scalar entries in vectors.)

36.

Let A be a $5 \times 3$ matrix, and let $\vec{y}$ be a vector in $\mbox{I\hspace{-.14em}R}^3$ and $\vec{z}$ a vector in $\mbox{I\hspace{-.14em}R}^5$. Suppose that $A\vec{y}=\vec{z}$. What fact allows you to conclude that the system $A\vec{x}=4\vec{z}$ is consistent?