- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.12 Exercises
$\quad$ In all exercises involving the vector space $V_n,$ the usual basis of unit coordinate vectors is to be chosen, unless another basis is specifically mentioned. In exercises concerned with the matrix of a linear transformation $T: V \rightarrow W$ where $V = W,$ we take the same basis in both $V$ and $W$ unless another choice is indicated.
4. $\quad$ A linear transformation $T: V_2 \rightarrow V_2$ is defined as follows: Each vector $(x, y)$ is reflected in the $y$-axis and then doubled in length to yield $T(x, y).$ Determine the matrix of $T$ and of $T^2.$
Solution. $\quad$ With unit coordinate basis elements $e_1 = w_1 = (1, 0)$ and $e_2 = w_2 = (0, 1),$ the transformation $T(x, y)$ for any $(x, y)$ in $V_2$ is given by: \begin{align*} T(x, y) &= 2(-x, y) \end{align*} Taking the transformation of the basis elements gives us: \begin{align*} T(e_1) &= T(1, 0) \\ &= (-2, 0) \\ &= -2w_1 + 0w_2 \\ \\ T(e_2) &= T(0, 1) \\ &= (0, 2) \\ &= 0w_1 + 2w_2 \end{align*} This gives us the matrix representation of $T:$ \begin{align*} (t_{ik}) &= \begin{bmatrix} -2 & 0 \\ 0 & 2 \end{bmatrix} \end{align*} Applying $T$ once more gives us: \begin{align*} \\ T^2(e_1) &= T[T(e_1)] \\ &= T(-2, 0) \\ &= (4, 0) \\ &= 4u_1 + 0u_2 \\ \\ T^2(e_2) &= T(0, 2) \\ &= (0, 4) \\ &= 0u_1 + 4u_2 \end{align*} where $u_1$ and $u_2$ are the unit coordinate basis elements of $T^2(V).$ This gives us the following matrix representation for $T^2:$ \begin{align*} (v_{ik}) &= \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix} \quad \blacksquare \end{align*}