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Let T: M2x2 → P₂ be defined by T A) A basis for the image (range) of T would be: {[][]} O{1, 1, 1, 2²} {[33]} o{0} O{1, x, x²} Ⓒ{1, 2²} {[89]} B) A basis for the kernel of T would be: {[3]} o {0} {[89]} O{1, ²} O{1, 1, 1, ²} {} =a+b+c+dx².

Let T: M2x2 → P₂ be defined by T A) A basis for the image (range) of T would be: {[][]} O{1, 1, 1, 2²} {[33]} o{0} O{1, x, x²} Ⓒ{1, 2²} {[89]} B) A basis for the kernel of T would be: {[3]} o {0} {[89]} O{1, ²} O{1, 1, 1, ²} {} =a+b+c+dx².

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Linear algebra
Let \( T: M_{2 \times 2} \rightarrow P_2 \) be defined by \( T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = a + b + c + dx^2 \). ### A) A basis for the image (range) of \( T \) would be: - \(\begin{Bmatrix} \begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix} \end{Bmatrix}\) - \(\{1, 1, 1, x^2\}\) - \(\begin{Bmatrix} \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} \end{Bmatrix}\) - \(\{0\}\) - \(\{1, x, x^2\}\) - **\(\{1, x^2\}\)** (Correct Answer) - \(\begin{Bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{Bmatrix}\) ### B) A basis for the kernel of \( T \) would be: - \(\begin{Bmatrix} \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} \end{Bmatrix}\) - \(\{0\}\) - \(\begin{Bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \end{Bmatrix}\) - \(\{1, x^2\}\) - \(\{1, 1, 1, x^2\}\) - **\(\{1, x, x^2\}\)** (Correct Answer) - \(\begin{Bmatrix} \begin{bmatrix} 1 & -1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ -1 & 0 \end{bmatrix} \end{Bmatrix}\)
Transcribed Image Text:Let \( T: M_{2 \times 2} \rightarrow P_2 \) be defined by \( T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = a + b + c + dx^2 \). ### A) A basis for the image (range) of \( T \) would be: - \(\begin{Bmatrix} \begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix} \end{Bmatrix}\) - \(\{1, 1, 1, x^2\}\) - \(\begin{Bmatrix} \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} \end{Bmatrix}\) - \(\{0\}\) - \(\{1, x, x^2\}\) - **\(\{1, x^2\}\)** (Correct Answer) - \(\begin{Bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{Bmatrix}\) ### B) A basis for the kernel of \( T \) would be: - \(\begin{Bmatrix} \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} \end{Bmatrix}\) - \(\{0\}\) - \(\begin{Bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \end{Bmatrix}\) - \(\{1, x^2\}\) - \(\{1, 1, 1, x^2\}\) - **\(\{1, x, x^2\}\)** (Correct Answer) - \(\begin{Bmatrix} \begin{bmatrix} 1 & -1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ -1 & 0 \end{bmatrix} \end{Bmatrix}\)
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