Represent a matrix on the (x,y)-plane

[hgj]

I'm having trouble understanding how to represent a matrix on the (x,y)-plane. It seems like my classes expect me to know this, but I was never shown how. I've heard that there are a lot of connections between matrices and geometry, that matrices can be used to provide a geometric representation of something, but I don't understand how. For example, if I have a 2x2 matrix $$\left(\begin{array}{cc}x&y\\0&1\end{array}\right)$$, how does that look in the (x,y)-plane? I really want to understand this better.

 

[amcavoy]

I'm having trouble understanding how to represent a matrix on the (x,y)-plane. It seems like my classes expect me to know this, but I was never shown how. I've heard that there are a lot of connections between matrices and geometry, that matrices can be used to provide a geometric representation of something, but I don't understand how. For example, if I have a 2x2 matrix $$\left(\begin{array}{cc}x&y\\0&1\end{array}\right)$$, how does that look in the (x,y)-plane? I really want to understand this better.

Well you could look at it as two vectors.

 

[HallsofIvy]

I have no idea what you mean by "represent a matrix on the xy-plane".

Perhaps you mean "represent a matrix as a linear transformation on the xy-plane.

If you are going to do that, it would be better not to have "x" and "y" in the matrix itself.
If the matrix were $$\left(\begin{array}{cc}2&3\\0&1\end{array}\right)$$
for example then we could note that multiplying the "point" (x,y) by it gives
$$\left(\begin{array}{cc}2&3\\0&1\end{array}\right)\left(\begin{array}{cc}x\\y\end{array}\right)= \left(\begin{array}{cc}2x+3y\\y\end{array}\right)$$

You could also, perhaps more simply, apply it to the "basis" vectors (1, 0) and (0,1) and see what happens there:

$$\left(\begin{array}{cc}2&3\\0&1\end{array}\right)\left(\begin{array}{cc}1\\0\end{array}\right)= \left(\begin{array}{cc}2\\0\end{array}\right)$$

$$\left(\begin{array}{cc}2&3\\0&1\end{array}\right)\left(\begin{array}{cc}0\\1\end{array}\right)= \left(\begin{array}{cc}3\\1\end{array}\right)$$

Draw the lines through (0,0) and each of those and imagine them as what the matrix does to the xy-axes. Of course, all points between the xy-axes are changed into points between those lines.