5.3E: Exercises for Section 5.3

Exercise \(\PageIndex{1}\)

Suppose that \(T :\mathbb{R}^n → \mathbb{R}^m\) is a linear transformation. Show that \(T(\vec{0})=\vec{0}\).

Exercise \(\PageIndex{2}\)

Let \(T\) be a linear transformation induced by the matrix \(A=\left[\begin{array}{cc}3&1\\-1&2\end{array}\right]\) and \(S\) a linear transformation induced by \(B=\left[\begin{array}{cc}0&-2\\4&2\end{array}\right]\). Find matrix of \(S\circ T\) and find \((S\circ T)(\vec{x})\) for \(\vec{x}=\left[\begin{array}{r}2\\-1\end{array}\right]\).

Answer

The matrix of \(S\circ T\) is given by \(BA\). \[\left[\begin{array}{cc}0&-2\\4&2\end{array}\right]\left[\begin{array}{cc}3&1\\-1&2\end{array}\right]=\left[\begin{array}{cc}2&-4\\10&8\end{array}\right]\nonumber\] Now, \((S\circ T)(\vec{x})=(BA)\vec{x}\). \[\left[\begin{array}{cc}2&-4\\10&8\end{array}\right]\left[\begin{array}{c}2\\-1\end{array}\right]=\left[\begin{array}{c}8\\12\end{array}\right]\nonumber\]

Exercise \(\PageIndex{3}\)

Let \(T\) be a linear transformation and suppose \(T\left(\left[\begin{array}{r}1\\-4\end{array}\right]\right)=\left[\begin{array}{r}2\\-3\end{array}\right]\). Suppose \(S\) is a linear transformation induced by the matrix \(B=\left[\begin{array}{cc}1&2\\-1&3\end{array}\right]\). Find \((S\circ T)(\vec{x})\) for \(\vec{x}=\left[\begin{array}{r}1\\-4\end{array}\right]\).

Answer

To find \((S\circ T)(\vec{x})\) we compute \(S(T(\vec{x}))\). \[\left[\begin{array}{cc}1&2\\-1&3\end{array}\right]\left[\begin{array}{c}2\\-3\end{array}\right]=\left[\begin{array}{c}-4\\-11\end{array}\right]\nonumber\]

Exercise \(\PageIndex{4}\)

Let \(T\) be a linear transformation induced by the matrix \(A=\left[\begin{array}{cc}2&3\\1&1\end{array}\right]\) and \(S\) a linear transformation induced by \(B=\left[\begin{array}{cc}-1&3\\1&-2\end{array}\right]\). Find matrix of \(S\circ T\) and find \((S\circ T)(\vec{x})\) for \(\vec{x}=\left[\begin{array}{c}5\\6\end{array}\right]\).

Exercise \(\PageIndex{5}\)

Let \(T\) be a linear transformation induced by the matrix \(A=\left[\begin{array}{cc}2&1\\5&2\end{array}\right]\). Find the matrix of \(T^{-1}\).

Answer

The matrix of \(T^{-1}\) is \(A^{-1}\). \[\left[\begin{array}{cc}2&1\\5&2\end{array}\right]^{-1}=\left[\begin{array}{cc}-2&1\\5&-2\end{array}\right]\nonumber\]

Exercise \(\PageIndex{6}\)

Let \(T\) be a linear transformation induced by the matrix \(A=\left[\begin{array}{cc}4&-3\\2&-2\end{array}\right]\). Find the matrix of \(T^{-1}\).

Exercise \(\PageIndex{7}\)

Let \(T\) be a linear transformation and suppose \(T\left(\left[\begin{array}{c}1\\2\end{array}\right]\right)=\left[\begin{array}{c}9\\8\end{array}\right],\:T\left(\left[\begin{array}{r}0\\-1\end{array}\right]\right)=\left[\begin{array}{c}-4\\-3\end{array}\right]\). Find the matrix of \(T^{-1}\).

Exercise \(\PageIndex{8}\)

Let \(V\) and \(W\) be subspaces of \(\mathbb{R}^n\) and \(\mathbb{R}^n\) respectively, and let \(T:V \to W\) be a linear transformation. Suppose that \( \{ T(\vec v_1), \cdots , T(\vec v_1) \}\) is a linearly independent set. Show that it must be the case that \( \{\vec v_1, \cdots , \vec v_n \} \) is also linearly independent.

Hint

Start with \( \sum_{i=1}^{r}a_i\vec v_i = 0 \). [We want to show that \( a_1 = \cdots = a_r = 0.\)] Use the properties of a linear transformation and the fact that \( \{T(\vec v_1), \cdots , T(\vec v_1) \}\) is a linearly independent set to finish the argument.