MATH221 April 2013
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Question 03

There are two linear transformations of $R^{2}$ that map the square with vertices at $(0,0),(1,0),(0,1),(1,1)$ to the square with vertices at $(0,0),(1,1),(1,1),(0,2)$. Call them $T_{1}$ and $T_{2}$. Determine the matrices of $T_{1}$ and $T_{2}$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

As the matrices must map the square edges to the square edges, we need to figure out where the four corners are mapped to.
Use the fact that T_{1} and T_{2} are linear, which restricts your options.

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Solution

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As that matrices map the square to the square, it will map the corners to the corners, thus we will see all possible ways to map the four corners to the four new corners.
 First, linear transforms must bring the vector $(0,0)$ to the vector $(0,0)$. So one of the corners is taken care of already.
 Next, note that there is a relation between the corners of the original square, that is $(1,0)+(0,1)=(1,1)$. As linear transformation preserve sums, T(a+b) = T(a)+T(b), we need to have the same relation in the target corners, which we do $(1,1)+(1,1)=(0,2)$. Hence any linear transform between the two squares must send (1,1) to (0,2).
 Thus, we are left with sending $(0,1),(1,0)$ to $(1,1),(1,1)$, and there are two ways to do so. As $(0,1),(1,0)$ forms a basis of the matrix, knowing where theses two vectors go also pins down the two matrices uniquely. We simply need to put the result in the columns of the transformation matrix:
$\left[{\begin{array}{cc}1&1\\1&1\end{array}}\right]$ and $\left[{\begin{array}{cc}1&1\\1&1\end{array}}\right]$


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