Understanding Vector Transformations: Problem 1

[jellicorse]

Hi,

I am trying to follow an introductory problem in my book for which no solutions are provided and have got stuck. I was wondering whether anyone could tell me how to go about this problem and where I am going wrong.

The problem starts:

Consider the eqquations:
$$y_1= x_1+2x_2$$
$$y_2=3x_2$$

We can view these equations as describing a transformation of the vector x = $\begin{bmatrix}x_1\\x_2\end{bmatrix}$ into the vector y = $\begin{bmatrix}y_1\\y_2\end{bmatrix}$

The transformation can be re-written as:

$$\begin{bmatrix}y_1\\y_2\end{bmatrix}=\begin{bmatrix}1 & 2\\0 & 3\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$

Or, more succinctly, y=Fx


Problem 1: Compute Fx For the following vectors x:

a) x=$\begin{bmatrix}1\\1\end{bmatrix}$ b) x=$\begin{bmatrix}1\\-1\end{bmatrix}$ c) x=$\begin{bmatrix}-1\\-1\end{bmatrix}$ d) x=$\begin{bmatrix}-1\\1\end{bmatrix}$


My Results:

a) Fx=$\begin{bmatrix}3\\3\end{bmatrix}$ b) Fx=$\begin{bmatrix}-1\\-3\end{bmatrix}$ c) Fx=$\begin{bmatrix}-3\\-3\end{bmatrix}$ a) Fx=$\begin{bmatrix}-1\\3\end{bmatrix}$


This is where I am unsure. The next step says "The heads of the four vectors x in problem 1 locate the four corners of a square in the $x_1x_2$ plane."

I'm not sure I understand this: what does the " $x_1x_2$ plane" mean? I would have thought it means a plane in which $x_1$ and $x_2$ are the axes... But I can't see how this can work as $x_1$ just consists of the points 3, -1,-3 and 1 on the x axis, as far as I can see...

I'd be very grateful if anyone could indicate where I'm going wrong..!

 

[chogg]

$x_1$ and $x_2$ refer to your original vectors. The 4 points are $(\pm 1, \pm 1)$; these are the corners of a square.

 

[jellicorse]

OK, thanks Chogg...