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

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In summary, you can think of a matrix as a way to transform points in space. It does this by affecting the axes that divide the space into coordinate systems.
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hgj
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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 [tex]\left(\begin{array}{cc}x&y\\0&1\end{array}\right)[/tex], how does that look in the (x,y)-plane? I really want to understand this better.
 
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  • #2
hgj said:
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 [tex]\left(\begin{array}{cc}x&y\\0&1\end{array}\right)[/tex], 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.
 
  • #3
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 [tex]\left(\begin{array}{cc}2&3\\0&1\end{array}\right)[/tex]
for example then we could note that multiplying the "point" (x,y) by it gives
[tex]\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)[/tex]

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

[tex]\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)[/tex]

[tex]\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)[/tex]

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.
 

What is a matrix and how is it represented on the (x,y)-plane?

A matrix is a rectangular array of numbers or symbols that are arranged in rows and columns. On the (x,y)-plane, a matrix is represented as a grid with the numbers or symbols placed in specific positions within the grid.

How do I read a matrix on the (x,y)-plane?

When reading a matrix on the (x,y)-plane, you start at the top left corner and move across the first row, then move down to the next row and continue until you reach the bottom right corner. The numbers or symbols in each position indicate the value or variable at that specific point on the plane.

What is the significance of representing a matrix on the (x,y)-plane?

Representing a matrix on the (x,y)-plane allows for easier visualization and comprehension of the data or relationships within the matrix. It also allows for easier manipulation and calculation of the data within the matrix.

What is the difference between a row matrix and a column matrix?

A row matrix is a matrix with only one row, while a column matrix is a matrix with only one column. In terms of representation on the (x,y)-plane, a row matrix would be a single line of numbers or symbols on the horizontal axis, while a column matrix would be a single line on the vertical axis.

Can a matrix be represented on other planes besides the (x,y)-plane?

Yes, a matrix can be represented on other planes, such as the (x,z)-plane or the (y,z)-plane. The specific plane used may depend on the context and purpose of the matrix.

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