Let \( T: M_{2 \times 2} \rightarrow P_2 \) be defined by \[ T \left( \begin{bmatrix} u & v \\ c & d \end{bmatrix} \right) = a + b + c + dx^2. \] ### A) A basis for the image (range) of \( T \) would be: - \(\begin{bmatrix} 2 & -1 \\ -1 & 0 \end{bmatrix}\) - \(\{0\}\) - \(\{1, 1, 1, x^2\}\) - \(\begin{bmatrix} 0 & 0 \end{bmatrix}\) - \(\{1, x^2\}\) - \(\{1, x, x^2\}\) - \(\begin{bmatrix} 1 & -1 & \begin{bmatrix} 1 & 0 \end{bmatrix} \\ 0 & 0 & \begin{bmatrix} -1 & 0 \end{bmatrix} \end{bmatrix}\) ### B) A basis for the kernel of \( T \) would be: - \(\begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix}\) - \(\begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}\) - \(\{1, x, x^2\}\) - \(\begin{bmatrix} 0 & 0 \end{bmatrix}\) - \(\{1, x^2\}\) - \(\{1, 1, 1, x^2\}\) - \(\{0\}\) This text represents a selection question commonly found in a linear algebra course, where students are tasked with identifying the basis of the image (range) and kernel of a given transformation \( T \).
Let \( T: M_{2 \times 2} \rightarrow P_2 \) be defined by \[ T \left( \begin{bmatrix} u & v \\ c & d \end{bmatrix} \right) = a + b + c + dx^2. \] ### A) A basis for the image (range) of \( T \) would be: - \(\begin{bmatrix} 2 & -1 \\ -1 & 0 \end{bmatrix}\) - \(\{0\}\) - \(\{1, 1, 1, x^2\}\) - \(\begin{bmatrix} 0 & 0 \end{bmatrix}\) - \(\{1, x^2\}\) - \(\{1, x, x^2\}\) - \(\begin{bmatrix} 1 & -1 & \begin{bmatrix} 1 & 0 \end{bmatrix} \\ 0 & 0 & \begin{bmatrix} -1 & 0 \end{bmatrix} \end{bmatrix}\) ### B) A basis for the kernel of \( T \) would be: - \(\begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix}\) - \(\begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}\) - \(\{1, x, x^2\}\) - \(\begin{bmatrix} 0 & 0 \end{bmatrix}\) - \(\{1, x^2\}\) - \(\{1, 1, 1, x^2\}\) - \(\{0\}\) This text represents a selection question commonly found in a linear algebra course, where students are tasked with identifying the basis of the image (range) and kernel of a given transformation \( T \).
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![Let \( T: M_{2 \times 2} \rightarrow P_2 \) be defined by
\[
T \left( \begin{bmatrix} u & v \\ c & d \end{bmatrix} \right) = a + b + c + dx^2.
\]
### A) A basis for the image (range) of \( T \) would be:
- \(\begin{bmatrix} 2 & -1 \\ -1 & 0 \end{bmatrix}\)
- \(\{0\}\)
- \(\{1, 1, 1, x^2\}\)
- \(\begin{bmatrix} 0 & 0 \end{bmatrix}\)
- \(\{1, x^2\}\)
- \(\{1, x, x^2\}\)
- \(\begin{bmatrix} 1 & -1 & \begin{bmatrix} 1 & 0 \end{bmatrix} \\ 0 & 0 & \begin{bmatrix} -1 & 0 \end{bmatrix} \end{bmatrix}\)
### B) A basis for the kernel of \( T \) would be:
- \(\begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix}\)
- \(\begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}\)
- \(\{1, x, x^2\}\)
- \(\begin{bmatrix} 0 & 0 \end{bmatrix}\)
- \(\{1, x^2\}\)
- \(\{1, 1, 1, x^2\}\)
- \(\{0\}\)
This text represents a selection question commonly found in a linear algebra course, where students are tasked with identifying the basis of the image (range) and kernel of a given transformation \( T \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F281d4858-639b-46d7-b04a-81e69daefef8%2Ff88a4299-8765-443a-827a-4a9b32756c7b%2Fxlnvm07.jpeg&w=3840&q=75)
Transcribed Image Text:Let \( T: M_{2 \times 2} \rightarrow P_2 \) be defined by
\[
T \left( \begin{bmatrix} u & v \\ c & d \end{bmatrix} \right) = a + b + c + dx^2.
\]
### A) A basis for the image (range) of \( T \) would be:
- \(\begin{bmatrix} 2 & -1 \\ -1 & 0 \end{bmatrix}\)
- \(\{0\}\)
- \(\{1, 1, 1, x^2\}\)
- \(\begin{bmatrix} 0 & 0 \end{bmatrix}\)
- \(\{1, x^2\}\)
- \(\{1, x, x^2\}\)
- \(\begin{bmatrix} 1 & -1 & \begin{bmatrix} 1 & 0 \end{bmatrix} \\ 0 & 0 & \begin{bmatrix} -1 & 0 \end{bmatrix} \end{bmatrix}\)
### B) A basis for the kernel of \( T \) would be:
- \(\begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix}\)
- \(\begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}\)
- \(\{1, x, x^2\}\)
- \(\begin{bmatrix} 0 & 0 \end{bmatrix}\)
- \(\{1, x^2\}\)
- \(\{1, 1, 1, x^2\}\)
- \(\{0\}\)
This text represents a selection question commonly found in a linear algebra course, where students are tasked with identifying the basis of the image (range) and kernel of a given transformation \( T \).
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