Let B be the basis of ?² consisting of the vectors ? 3 ? ? -2 ? ? ? and ? ? , ? 1 ? ? 5 ? and let R be the basis consisting of ? 2 ? ? -3 ? ? ? and ? ? . ? -1 ? ? 2 ? Find a matrix P such that [x]R = P[x]B for all x in ?². P = ? ? ? ? ? ? ? ?
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To do this, we will express the vectors of R in terms of the basis B. Let the vectors in basis B be $\textbf{b}_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\textbf{b}_2 = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$. Let the vectors in basis R be $\textbf{r}_1 = Show more…
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Consider the basis $$\$$ of $\mathbb{R}^{2}$ consisting of the vectors $\left[\begin{array}{l}1 \\ 1\end{array}\right]$ and $\left[\begin{array}{l}1 \\ 2\end{array}\right],$ and let $\Re$ be the basis consisting of $\left[\begin{array}{l}1 \\ 2\end{array}\right]$ $\left[\begin{array}{l}3 \\ 4\end{array}\right] .$ Find a matrix $P$ such that \[ [\vec{x}]_{\Re}=P[\vec{x}]_{\mathfrak{B}} \] for all $\vec{x}$ in $\mathbb{R}^{2}.$
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Let $T: R^{2} \rightarrow R^{2}$ be a linear operator, and let $B$ and $B^{\prime}$ be bases for $R^{2}$ for which $$ [T]_{B^{\prime}}=\left[\begin{array}{rr} 3 & 2 \\ -1 & 1 \end{array}\right] \quad \text { and } \quad P_{B^{\prime} \rightarrow B}=\left[\begin{array}{rr} 4 & 5 \\ 1 & -1 \end{array}\right] $$ Find the matrix for $T$ relative to the basis $B$
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