1. ## Transformations

If T maps $\displaystyle \vec x$ into $\displaystyle \vec w$, and the standard matrix is defined by the equations:
$\displaystyle w_1=2x_1-3x_2+x_3$
$\displaystyle w_2=3x_1+5x_2-x_3$

How do I find the standard matrix? And then how do I find the image of vectors under T by substituting in the equations and using the standard matrix?

The prof didn't go over any of this stuff, and yet still assigned one question as homework, and unfortunately the text is rather vague...

2. Your T is such that

$\displaystyle T\vec x = \vec w$

So

$\displaystyle T=\left[\begin{array}{ccc}2 & -1 & 1\\ 3 & 5 & -1 \end{array}\right]$

Check this.

3. Alright so I kind of guessed that using some of the examples... But as for finding an image, all I would need to do is multiply the matrix you gave me with a matrix that substitutes for $\displaystyle x_1,x_2,$ and $\displaystyle x_3$ correct?

4. Yep For example, if x=(1,1,1) then Tx=(2,7)

$\displaystyle \left[\begin{array}{ccc}2 & -1 & 1\\ 3 & 5 & -1 \end{array}\right]\left[\begin{array}{c}1\\1\\1\end{array}\right]=\left[\begin{array}{c}2\\7\end{array}\right]$

5. Awesome, thanks so much.