Suppose T: R¹ R² is a linear transformation with T(e₁) [1]. Tez) [12]. Te T(₂) T(es) T(е4) T 9 5 -8 17 NOTE: e, refers to the ith column of the nx n identity matrix. Find T Check Answer 9 -8 8 6 H -[+]
Suppose T: R¹ R² is a linear transformation with T(e₁) [1]. Tez) [12]. Te T(₂) T(es) T(е4) T 9 5 -8 17 NOTE: e, refers to the ith column of the nx n identity matrix. Find T Check Answer 9 -8 8 6 H -[+]
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
![Suppose \( T : \mathbb{R}^4 \rightarrow \mathbb{R}^2 \) is a linear transformation with
\[
T(e_1) =
\begin{bmatrix}
2 \\
4
\end{bmatrix}
, \quad
T(e_2) =
\begin{bmatrix}
13 \\
7
\end{bmatrix}
, \quad
T(e_3) =
\begin{bmatrix}
4 \\
12
\end{bmatrix}
,
\]
\[
T(e_4) =
\begin{bmatrix}
-5 \\
17
\end{bmatrix}
.
\]
Find
\[
T
\begin{pmatrix}
9 \\
-8 \\
8 \\
6
\end{pmatrix}
.
\]
NOTE: \( e_i \) refers to the \( i^{th} \) column of the \( n \times n \) identity matrix.
\[
T
\begin{pmatrix}
9 \\
-8 \\
8 \\
6
\end{pmatrix}
=
\begin{bmatrix}
\phantom{-} \\
\phantom{-}
\end{bmatrix}
\]
[Check Answer]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18574973-f25e-4ab8-b7d6-6007b5b87fc4%2F0cca0af2-680a-4d97-bbb4-8c401d57ac36%2Fvrj7mto_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose \( T : \mathbb{R}^4 \rightarrow \mathbb{R}^2 \) is a linear transformation with
\[
T(e_1) =
\begin{bmatrix}
2 \\
4
\end{bmatrix}
, \quad
T(e_2) =
\begin{bmatrix}
13 \\
7
\end{bmatrix}
, \quad
T(e_3) =
\begin{bmatrix}
4 \\
12
\end{bmatrix}
,
\]
\[
T(e_4) =
\begin{bmatrix}
-5 \\
17
\end{bmatrix}
.
\]
Find
\[
T
\begin{pmatrix}
9 \\
-8 \\
8 \\
6
\end{pmatrix}
.
\]
NOTE: \( e_i \) refers to the \( i^{th} \) column of the \( n \times n \) identity matrix.
\[
T
\begin{pmatrix}
9 \\
-8 \\
8 \\
6
\end{pmatrix}
=
\begin{bmatrix}
\phantom{-} \\
\phantom{-}
\end{bmatrix}
\]
[Check Answer]
![Let \( S \) be a linear transformation from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \) with associated matrix
\[
A = \begin{bmatrix} 3 & 3 & -3 \\ 1 & 2 & -1 \end{bmatrix}.
\]
Let \( T \) be a linear transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \) with associated matrix
\[
B = \begin{bmatrix} 2 & -2 \\ 1 & 2 \end{bmatrix}.
\]
Determine the matrix \( C \) of the composition \( T \circ S \).
\[
C = \begin{bmatrix} \Box & \Box & \Box \\ \Box & \Box & \Box \end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18574973-f25e-4ab8-b7d6-6007b5b87fc4%2F0cca0af2-680a-4d97-bbb4-8c401d57ac36%2Fqpugcvc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let \( S \) be a linear transformation from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \) with associated matrix
\[
A = \begin{bmatrix} 3 & 3 & -3 \\ 1 & 2 & -1 \end{bmatrix}.
\]
Let \( T \) be a linear transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \) with associated matrix
\[
B = \begin{bmatrix} 2 & -2 \\ 1 & 2 \end{bmatrix}.
\]
Determine the matrix \( C \) of the composition \( T \circ S \).
\[
C = \begin{bmatrix} \Box & \Box & \Box \\ \Box & \Box & \Box \end{bmatrix}
\]
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