# Linear Combinations

• Sep 30th 2010, 11:28 PM
Ferny84
Linear Combinations
A = $$\left( \begin{array}{ccc} 2 & 3 & 1 \\ 3 & -1 & 4 \\ -1 & 0 & 1 \end{array} \right)$$,
x = $$\left( \begin{array}{ccc} 1 \\ -2 \\ 3 \end{array} \right)$$,
y = $$\left( \begin{array}{ccc} 3\\ 0\\ 1 \end{array} \right)$$,
z = $$\left( \begin{array}{ccc} 4\\ -2\\ 5 \end{array} \right)$$

I need to compute Ax, Ay, Az as linear combinations of the columns of A.

I'm stuck here, I have tried multiplying x, y and z with A and got:

x = $$\left( \begin{array}{ccc} -1 \\ 17 \\ 2 \end{array} \right)$$,
y = $$\left( \begin{array}{ccc} 7\\ 13\\ -2 \end{array} \right)$$,
z = $$\left( \begin{array}{ccc} 7\\ 34\\ 1 \end{array} \right)$$

Then it asks me to use the answers to compute the product:

$$\left( \begin{array}{ccc} 2 & 3 & 1 \\ 3 & -1 & 4 \\ -1 & 0 & 1 \end{array} \right)$$ $$\left( \begin{array}{ccc} 1 & 3 & 4 \\ -2 & 0 & -2 \\ 3 & 1 & 5 \end{array} \right)$$

Can someone help me out on what to do?

I'm really confused...
• Oct 1st 2010, 04:36 AM
HallsofIvy
First you should write those results as "Ax= ", "Ay= ", and "Az= ", not "x= ", etc.

The columns of A are, of course, $\begin{bmatrix}2 \\ 3\\ -1\end{bmatrix}$, $\begin{bmatrix}3 \\ -1\\ 0\end{bmatrix}$, and $\begin{bmatrix}1 \\ 4\\ 1\end{bmatrix}$

You want to write [tex]Ax= $\begin{bmatrix}-1 \\ 17\\ 2\end{bmatrix}= \alpha\begin{bmatrix}2 \\ 3\\ -1\end{bmatrix}+ \beta\begin{bmatrix}3 \\ -1\\ 0\end{bmatrix}+ \gamma\begin{bmatrix}1 \\ 4\\ 1\end{bmatrix}$ $= \begin{bmatrix}2\alpha+ 3\beta+ \gamma \\ 3\alpha- \beta+ 4\gamma \\ -\alpha+ \gamma\end{bmatrix}$
so you need to solve the three equations $2\alpha+ 3\beta+ \gamma= -1$, $3\alpha- \beta+ 4\gamma= 17$, and $-\alpha+ \gamma= 2$ for $\alpha$, $\beta$, and $\gamma$.

For the second problem, you are simply being asked to multiply two matrices. Don't you know how to do that? You seem to have multiplied a matrix and a vector (column matrix). Just multiply the first matrix times each column in the second matrix, giving the three columns of the product matrix.