Science:Math Exam Resources/Courses/MATH221/April 2013/Question 07 (b)

MATH221 April 2013

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Question 07 (b)
Let $T:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ be the linear transformation that reflects points through the line 3x = 4y. b) Using the result of a), find A (the standard matrix of T).

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Hint
Using the information from part a), the matrix of T can be described by using the eigenvectors and eigenvalues.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. What we know from part a) Using the eigenvectors $\left[{\begin{array}{c}4\\3\end{array}}\right],\left[{\begin{array}{c}-3\\4\end{array}}\right]$, and the eigenvalues $\ 1,-1$, we know that $A\left[{\begin{array}{c}4\\3\end{array}}\right]=\left[{\begin{array}{c}4\\3\end{array}}\right]$ and $A\left[{\begin{array}{c}-3\\4\end{array}}\right]=-\left[{\begin{array}{c}-3\\4\end{array}}\right]$. Finding the formula for the matrix $\ A$ Then, we can write the information from above in matrix notation. ${\begin{aligned}A\left[{\begin{array}{cc}4&-3\\3&4\end{array}}\right]&={\begin{bmatrix}A{\begin{bmatrix}4\\3\end{bmatrix}}\ A{\begin{bmatrix}-3\\4\end{bmatrix}}\end{bmatrix}}\\&={\begin{bmatrix}(1){\begin{bmatrix}4\\3\end{bmatrix}}\ (-1){\begin{bmatrix}-3\\4\end{bmatrix}}\end{bmatrix}}\\&=(1){\begin{bmatrix}4&0\\3&0\end{bmatrix}}+(-1){\begin{bmatrix}0&-3\\0&4\end{bmatrix}}\\&={\begin{bmatrix}4\\3\end{bmatrix}}{\begin{bmatrix}1&0\end{bmatrix}}+{\begin{bmatrix}-3\\4\end{bmatrix}}{\begin{bmatrix}0&-1\end{bmatrix}}\\&={\begin{bmatrix}4&-3\\3&4\end{bmatrix}}{\begin{bmatrix}1&0\\0&-1\end{bmatrix}}\end{aligned}}$ The first equality follows from this: when we product two 2 by 2 matrices, the first column of the resulting matrix is the product of the first matrix with the first column of the second matrix. Similarly, the second column of the resulting matrix is the product of the first matrix with the second column of the second matrix. In the last step we use, in reverse, that the product of two matrices is the sum of (columns of the first matrix) times (rows of the second matrix). We define the matrix $B={\begin{bmatrix}4&-3\\3&4\end{bmatrix}}$ as the matrix of eigenvectors of $A$ and write the equation above as $AB=B\left[{\begin{array}{cc}1&0\\0&-1\end{array}}\right].$ This leads us to $A=B\left[{\begin{array}{cc}1&0\\0&-1\end{array}}\right]B^{-1}$. Calculating $\ A$ First the inverse of B is given by the usual method for 2x2 matrices (where $25=\det(B)$): $B^{-1}={\frac {1}{25}}\left[{\begin{array}{cc}4&3\\-3&4\end{array}}\right]$ We can now compute A as claimed: $A=\left[{\begin{array}{cc}4&-3\\3&4\end{array}}\right]\left[{\begin{array}{cc}1&0\\0&-1\end{array}}\right]\left({\frac {1}{25}}\left[{\begin{array}{cc}4&3\\-3&4\end{array}}\right]\right)={\frac {1}{25}}\left[{\begin{array}{cc}7&24\\24&-7\end{array}}\right]$.

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