Question
Let $T: P_{2}(\mathbb{R}) \rightarrow \mathbb{R}^{3}$ be given by$$T(p(x))=(p(0), p(1), p(2))$$Is $T$ invertible? Find the matrix representation of $T$ with respect to the standard bases and use it to support your answer.
Step 1
To do this, we apply $T$ to each vector in the basis of $P_{2}(\mathbb{R})$, which is $\{1, x, x^2\}$, and express the result as a linear combination of the basis vectors of $\mathbb{R}^{3}$, which is $\{(1,0,0), (0,1,0), (0,0,1)\}$. Show more…
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Let $T: P_{2} \rightarrow P_{3}$ be the linear transformation defined by $T(p(x))=x p(x)$ (a) Find the matrix for $T$ relative to the standard bases $$B=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\} \quad \text { and } \quad B^{\prime}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{v}_{4}\right\}$$ where $$\begin{array}{lll} \mathbf{u}_{1}=1, & \mathbf{u}_{2}=x, & \mathbf{u}_{3}=x^{2} \\ \mathbf{v}_{1}=1, & \mathbf{v}_{2}=x, & \mathbf{v}_{3}=x^{2}, \quad \mathbf{v}_{4}=x^{3} \end{array}$$ (b) Verify that the matrix $[T]_{B^{\prime}, B}$ obtained in part (a) satisfies Formula (5) for every vector $\mathbf{x}=c_{0}+c_{1} x+c_{2} x^{2}$ in $P_{2}$
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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 $$ [T]_{B}=\left[\begin{array}{ll} 2 & 0 \\ 1 & 1 \end{array}\right] \quad \text { and } \quad P_{B \rightarrow B^{\prime}}=\left[\begin{array}{ll} 3 & 2 \\ 1 & 1 \end{array}\right] $$ Find the matrix for $T$ relative to the basis $B^{\prime}$.
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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 $$ [T]_{B^{\prime}}=\left[\begin{array}{ll} 2 & 0 \\ 1 & 1 \end{array}\right] \quad \text { and } \quad P_{B \rightarrow B^{\prime}}=\left[\begin{array}{ll} 3 & 2 \\ 1 & 1 \end{array}\right] $$ Find the matrix for $T$ relative to the basis $B$.
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