5.8E: Exercises for Section 5.8

Exercise \(\PageIndex{1}\)

Let \(B=\left\{\left[\begin{array}{r}2\\-1\end{array}\right],\:\left[\begin{array}{r}3\\2\end{array}\right]\right\}\) be a basis of \(\mathbb{R}^2\) and let \(\vec{x}=\left[\begin{array}{r}5\\-7\end{array}\right]\) be a vector in \(\mathbb{R}^2\). Find \(C_B(\vec{x})\).

Exercise \(\PageIndex{2}\)

Let \(B=\left\{\left[\begin{array}{r}1\\-1\\2\end{array}\right],\:\left[\begin{array}{r}2\\1\\2\end{array}\right],\:\left[\begin{array}{r}-1\\0\\2\end{array}\right]\right\}\) be a basis of \(\mathbb{R}^3\) and let \(\vec{x}=\left[\begin{array}{r}5\\-1\\4\end{array}\right]\) be a vector in \(\mathbb{R}^2\). Find \(C_B(\vec{x})\).

Answer

\(C_B(\vec{x})=\left[\begin{array}{r}2\\1\\-1\end{array}\right]\)

Exercise \(\PageIndex{3}\)

Let \(T\): \(\mathbb{R}^2→\mathbb{R}^2\) be a linear transformation defined by \(T\left(\left[\begin{array}{c}a\\b\end{array}\right]\right)=\left[\begin{array}{c}a+b\\a-b\end{array}\right]\).

Consider the two bases \[B_1=\{\vec{v}_1,\vec{v}_2\}=\left\{\left[\begin{array}{c}1\\0\end{array}\right],\:\left[\begin{array}{r}-1\\1\end{array}\right]\right\}\nonumber\] and \[B_2=\left\{\left[\begin{array}{c}1\\1\end{array}\right],\:\left[\begin{array}{r}1\\-1\end{array}\right]\right\}\nonumber\]

Find the matrix \(M_{B_2 B_1}\) of \(T\) with respect to the bases \(B_1\) and \(B_2\).

Answer

\(M_{B_2B_1}=\left[\begin{array}{rr}1&0\\-1&1\end{array}\right]\)E

Exercise \(\PageIndex{4}\)

Consider the basis \(B\) of \(\mathbb{R}^3\) given by \[B = \{\vec{v}_1 , \vec{v}_2, \vec{v}_3 \} = \left\{ \left [ \begin{array}{r} 2 \\ 1 \\ 1 \end{array} \right ] ,\left [ \begin{array}{r} 1 \\ 1 \\ 0 \end{array} \right ] ,\left [ \begin{array}{r} 0 \\ 0 \\ 1 \end{array} \right ] \right\}\nonumber \] And let \(T :\mathbb{R}^{3}\mapsto \mathbb{R}^{3}\) be the linear transformation defined on \(B\) as: \[T\left(\left [ \begin{array}{r} 2\\ 1 \\ 1 \end{array} \right ]\right) =\left [ \begin{array}{r} 1 \\ 2 \\ -1 \end{array} \right ] ,T \left(\left [ \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right ]\right) =\left [ \begin{array}{r} 1 \\ 3 \\ 1 \end{array} \right ] ,T\left(\left [ \begin{array}{r} 0 \\ 0 \\ 1 \end{array} \right ] \right)=\left [ \begin{array}{r} 0 \\ 2 \\ 2 \end{array} \right ]\nonumber \]

1. Find the matrix \(M_{B}\) of \(T\) relative to the basis \(B\).
2. Then find the usual matrix of \(T\) with respect to the standard basis of \(\mathbb{R}^{3}\).

Exercise \(\PageIndex{5}\)

Prove that for any basis \(B\) of \(\mathbb{R}^n\), the coordinate function \[C_B: \mathbb{R}^n \rightarrow \mathbb{R}^n\nonumber \] is a linear transformation.