Consider the vectors u₁ = ------- 2 and the vector : 3 B = If IS in the span of ₁, ₂, 3, 4, write was a linear combination of 1, 2, 3, 4, using as few nonzero coefficients as possible. X U2+ 1 3 →→→ If w IS NOT in the span of u₁, 2, 3, 4, write DNE in each of the answer boxes. w = 7 x uit- XU3+ 2 X U4
Consider the vectors u₁ = ------- 2 and the vector : 3 B = If IS in the span of ₁, ₂, 3, 4, write was a linear combination of 1, 2, 3, 4, using as few nonzero coefficients as possible. X U2+ 1 3 →→→ If w IS NOT in the span of u₁, 2, 3, 4, write DNE in each of the answer boxes. w = 7 x uit- XU3+ 2 X U4
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
![**Vectors and Linear Combinations**
Consider the vectors:
\[
\vec{u_1} = \begin{bmatrix} 1 \\ 1 \\ -2 \end{bmatrix}, \quad \vec{u_2} = \begin{bmatrix} 1 \\ 2 \\ -2 \end{bmatrix}, \quad \vec{u_3} = \begin{bmatrix} 1 \\ -2 \\ -2 \end{bmatrix}, \quad \vec{u_4} = \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix}
\]
and the vector:
\[
\vec{w} = \begin{bmatrix} 3 \\ -3 \\ -6 \end{bmatrix}.
\]
### Instructions:
**If** \(\vec{w}\) **is in the span of** \(\vec{u_1}, \vec{u_2}, \vec{u_3}, \vec{u_4}\), write \(\vec{w}\) as a linear combination of \(\vec{u_1}, \vec{u_2}, \vec{u_3}, \vec{u_4}\), **using as few nonzero coefficients as possible**.
**If** \(\vec{w}\) **is NOT in the span of** \(\vec{u_1}, \vec{u_2}, \vec{u_3}, \vec{u_4}\), write "DNE" in each of the answer boxes.
### Solution:
\[
\vec{w} = 7 \vec{u_1} + (-7) \vec{u_2} + 1 \vec{u_3} + 2 \vec{u_4}
\]
This expression indicates that \(\vec{w}\) can be expressed as a linear combination of the vectors \(\vec{u_1}, \vec{u_2}, \vec{u_3}, \vec{u_4}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c49f70f-02c0-4076-b0a5-564b9a6f1d8d%2F54ba86b3-3aae-4c5d-8c02-b575c6ccb791%2Fjx5hdy_processed.png&w=3840&q=75)
Transcribed Image Text:**Vectors and Linear Combinations**
Consider the vectors:
\[
\vec{u_1} = \begin{bmatrix} 1 \\ 1 \\ -2 \end{bmatrix}, \quad \vec{u_2} = \begin{bmatrix} 1 \\ 2 \\ -2 \end{bmatrix}, \quad \vec{u_3} = \begin{bmatrix} 1 \\ -2 \\ -2 \end{bmatrix}, \quad \vec{u_4} = \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix}
\]
and the vector:
\[
\vec{w} = \begin{bmatrix} 3 \\ -3 \\ -6 \end{bmatrix}.
\]
### Instructions:
**If** \(\vec{w}\) **is in the span of** \(\vec{u_1}, \vec{u_2}, \vec{u_3}, \vec{u_4}\), write \(\vec{w}\) as a linear combination of \(\vec{u_1}, \vec{u_2}, \vec{u_3}, \vec{u_4}\), **using as few nonzero coefficients as possible**.
**If** \(\vec{w}\) **is NOT in the span of** \(\vec{u_1}, \vec{u_2}, \vec{u_3}, \vec{u_4}\), write "DNE" in each of the answer boxes.
### Solution:
\[
\vec{w} = 7 \vec{u_1} + (-7) \vec{u_2} + 1 \vec{u_3} + 2 \vec{u_4}
\]
This expression indicates that \(\vec{w}\) can be expressed as a linear combination of the vectors \(\vec{u_1}, \vec{u_2}, \vec{u_3}, \vec{u_4}\).
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