convert values from one coordinate system (x,y) to another coordinate system (x', y')

Following is a graph that contains both coordinate systems (x,y) and (x',y'). x, y, x', and y' are all axes

y'
^
|
+----------> x
|  |  |  |
+--+--+--+
|  |  |  |
+--+--+--+
|  |  |  |
+--+--+--+-> x'
|
v
y


Above is a picture of a graph. For example, the following is graph or grid of the (x',y') coordinate system.

y'
^
|
+--------+
|  |  |  |
+--+--+--+
|  |  |  |
+--+--+--+
|  |  |  |
+--+--+--+-> x'


What is the transformation T which transforms coordinates (x,y) to coordinates (x',y')? Please include simple, intuitive explanations and steps. Note: the solution, T, is probably a 2d matrix.

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Your $x'$ is the same as your old $x$. Your $y'$ is $3-y$. – user17762 Jun 21 '12 at 22:16

Note that your $x'$ is the same as your $x$. To get $y'$, first note that $y$ is in the opposite direction of $y'$. Hence, $y = c - y'$ where $c$ is some constant. The constant is obtained by finding the value of $y$ at say $y'=0$, in your case the constant is $3$.
It looks to me like you will have $x=x'$ and $y=3-y'$.
The solution is not going to be a matrix, because the transformation is not linear. Linear transformations always preserve the origin of the coordinate system, and in your diagram the origin for the $x$-$y$ axes is in a different place than the origin for the $x'$-$y'$ axes.