We are given the following vectors in R³: 2 1 Let T: R³ → R2 be a linear transformation such that W1 Find the vector such that T (v) = X1 X2 Hint. Express V as a linear combination of W₁, W₂, and W3 first. Enter your answer in the form [x₁, x₂]: 3 W₂ = X2 [] [3] W3 = T(W₁) = T(W₂) = T (W3) V= = [2] 3
We are given the following vectors in R³: 2 1 Let T: R³ → R2 be a linear transformation such that W1 Find the vector such that T (v) = X1 X2 Hint. Express V as a linear combination of W₁, W₂, and W3 first. Enter your answer in the form [x₁, x₂]: 3 W₂ = X2 [] [3] W3 = T(W₁) = T(W₂) = T (W3) V= = [2] 3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![We are given the following vectors in \( \mathbb{R}^3 \):
\[
\mathbf{w_1} = \begin{bmatrix} 2 \\ 1 \\ 4 \end{bmatrix}, \quad \mathbf{w_2} = \begin{bmatrix} 4 \\ -3 \\ 3 \end{bmatrix}, \quad \mathbf{w_3} = \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} -4 \\ 3 \\ -3 \end{bmatrix}
\]
Let \( T : \mathbb{R}^3 \to \mathbb{R}^2 \) be a linear transformation such that
\[
T(\mathbf{w_1}) = T(\mathbf{w_2}) = T(\mathbf{w_3}) = \begin{bmatrix} 3 \\ 2 \end{bmatrix}
\]
Find the vector \(\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\) such that \( T(\mathbf{v}) = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \).
**Hint.** Express \(\mathbf{v}\) as a linear combination of \(\mathbf{w_1}, \mathbf{w_2},\) and \(\mathbf{w_3}\) first.
Enter your answer in the form \([x_1, x_2]\):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb493ecdd-cbdc-400d-a05a-de2010eb2d52%2F5e3df663-c882-4e92-afba-b7b117367653%2F0ze8znr_processed.png&w=3840&q=75)
Transcribed Image Text:We are given the following vectors in \( \mathbb{R}^3 \):
\[
\mathbf{w_1} = \begin{bmatrix} 2 \\ 1 \\ 4 \end{bmatrix}, \quad \mathbf{w_2} = \begin{bmatrix} 4 \\ -3 \\ 3 \end{bmatrix}, \quad \mathbf{w_3} = \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} -4 \\ 3 \\ -3 \end{bmatrix}
\]
Let \( T : \mathbb{R}^3 \to \mathbb{R}^2 \) be a linear transformation such that
\[
T(\mathbf{w_1}) = T(\mathbf{w_2}) = T(\mathbf{w_3}) = \begin{bmatrix} 3 \\ 2 \end{bmatrix}
\]
Find the vector \(\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\) such that \( T(\mathbf{v}) = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \).
**Hint.** Express \(\mathbf{v}\) as a linear combination of \(\mathbf{w_1}, \mathbf{w_2},\) and \(\mathbf{w_3}\) first.
Enter your answer in the form \([x_1, x_2]\):
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