1. Let T:R³ a T b с = R4 be defined by a + 2b - c 3a + 4b 2a - 3c 56 Sh
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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Question
Find the basis for both Range(T) and Ker(T).
![1. Let \( T: \mathbb{R}^3 \to \mathbb{R}^4 \) be defined by
\[
T \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} a + 2b - c \\ 3a + 4b \\ 2a - 3c \\ 5b \end{pmatrix}.
\]
This expression defines a linear transformation \( T \) that maps vectors from \(\mathbb{R}^3\) to \(\mathbb{R}^4\). Here, a vector \(\begin{pmatrix} a \\ b \\ c \end{pmatrix}\) in \(\mathbb{R}^3\) is transformed into a vector \(\begin{pmatrix} a + 2b - c \\ 3a + 4b \\ 2a - 3c \\ 5b \end{pmatrix}\) in \(\mathbb{R}^4\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd11102b1-e9d2-4b8f-a3bc-cb638b777a97%2Ffd733fa7-251b-4681-ad08-4109d0b9d00e%2F8m1u9v_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let \( T: \mathbb{R}^3 \to \mathbb{R}^4 \) be defined by
\[
T \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} a + 2b - c \\ 3a + 4b \\ 2a - 3c \\ 5b \end{pmatrix}.
\]
This expression defines a linear transformation \( T \) that maps vectors from \(\mathbb{R}^3\) to \(\mathbb{R}^4\). Here, a vector \(\begin{pmatrix} a \\ b \\ c \end{pmatrix}\) in \(\mathbb{R}^3\) is transformed into a vector \(\begin{pmatrix} a + 2b - c \\ 3a + 4b \\ 2a - 3c \\ 5b \end{pmatrix}\) in \(\mathbb{R}^4\).
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