Question 1.7: Find the standard matrix for the linear operator T : R² → R²......
Find the standard matrix for the linear operator T : R² → R² that is a dilation of a vector with factor k = 2, then the resulting vector is rotated by θ = 45°, and then contraction is applied with factor k = \sqrt{2}.
The linear operator T can be expressed as the composition \scriptstyle T\;=\;T_{3}\circ T_{2}\circ T_{1}, where T_{1} is the dilation with factor k = 2, T_{2} is the rotation with θ = 45°, and T_{3} is the contraction with factor k = \sqrt{2}. If the standard matrices for these linear transformations are
\scriptstyle T_{1}={\left[\begin{array}{l l}{k}&{0}\\ {0}&{k}\end{array}\right]} with k = 2, then T_{1}=\left[\begin{array}{c c}{{2}}&{{0}}\\ {{0}}&{{2}}\end{array}\right]
T_{2}={\left[\begin{array}{l l}{\ \cos\theta}&{-\sin\theta \ }\\ {\ \sin\theta}&{\ \ \cos\theta}\end{array}\right]}with θ = 45°, thenT_{\mathrm{2}}=\!\left[\begin{array}{c c}{{{\frac{1}{\sqrt{2}}}}}&{{-{\frac{1}{\sqrt{2}}}}}\\ {{{\frac{1}{\sqrt{2}}}}}&{{{\frac{1}{\sqrt{2}}}}}\end{array}\right]
T_{\mathrm{3}}=\left[\begin{array}{c c}{{k}}&{{0}}\\ {{0}}&{{k}}\end{array}\right] with k = \sqrt{2}, then T_{3}={\left[\begin{array}{l l}{{\sqrt{2}}}&{\ \ 0}\\ {\ \ 0}&{{\sqrt{2}}}\end{array}\right]}.
Hence, \scriptstyle T\;=\;T_{3}\circ T_{2}\circ T_{1}=T_{3}\cdot T_{2}\cdot T_{1} =\textstyle{\left[\begin{array}{l l}{2}&{0}\\ {0}&{2}\end{array}\right]}\cdot \left[\begin{array}{l l}{{\frac{1}{\sqrt{2}}}}&{-{\frac{1}{\sqrt{2}}}}\\ {\frac{1}{\sqrt{2}}}&{{\ \ \ \frac{1}{\sqrt{2}}}}\end{array}\right]\cdot \left[\begin{array}{l l}{{\sqrt{2}}}&{\ \ 0}\\ {\ \ 0}&{{\sqrt{2}}}\end{array}\right] .
∴T=T_3\circ T_2 \circ T= \left[{\begin{array}{l l}{2}&{-2}\\ {2}&{\ \ 2}\end{array}}\right].