13. 1 H a. Is w in {V1, V2, V3}? How many vectors are in {V₁, V2, V3}? b. How many vectors are in Span {V₁, V2, V3}? c. Is w in the subspace spanned by {V₁, V2, V3}? Why? Let V1 = , V2 2 G. = V3 2 and w = 3 O 1 2
13. 1 H a. Is w in {V1, V2, V3}? How many vectors are in {V₁, V2, V3}? b. How many vectors are in Span {V₁, V2, V3}? c. Is w in the subspace spanned by {V₁, V2, V3}? Why? Let V1 = , V2 2 G. = V3 2 and w = 3 O 1 2
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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![**Problem 13**
Given:
\[ \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 4 \\ 2 \\ 6 \end{bmatrix}, \quad \text{and} \quad \mathbf{w} = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}. \]
### (a) Is \(\mathbf{w}\) in \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\)? How many vectors are in \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\)?
The set \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\) is asked to be examined to see if it contains the vector \(\mathbf{w}\).
### (b) How many vectors are in Span \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\)?
Determine the number of vectors in the span of \(\mathbf{v}_1, \mathbf{v}_2\), and \(\mathbf{v}_3\), meaning the set of all possible linear combinations of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\).
### (c) Is \(\mathbf{w}\) in the subspace spanned by \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\)? Why?
Examine whether the vector \(\mathbf{w}\) can be written as a linear combination of the vectors \(\mathbf{v}_1, \mathbf{v}_2\), and \(\mathbf{v}_3\). In other words, explore if there exist scalars \(a, b, c\) such that \( a\mathbf{v}_1 + b\mathbf{v}_2 + c\mathbf{v}_3 = \mathbf{w} \). If so,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324f16c4-c4e5-4b06-b074-f261b39e021a%2Fb2b2399d-51b9-4cb0-a54c-f105c1554a03%2Flkmi0xq_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 13**
Given:
\[ \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 4 \\ 2 \\ 6 \end{bmatrix}, \quad \text{and} \quad \mathbf{w} = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}. \]
### (a) Is \(\mathbf{w}\) in \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\)? How many vectors are in \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\)?
The set \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\) is asked to be examined to see if it contains the vector \(\mathbf{w}\).
### (b) How many vectors are in Span \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\)?
Determine the number of vectors in the span of \(\mathbf{v}_1, \mathbf{v}_2\), and \(\mathbf{v}_3\), meaning the set of all possible linear combinations of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\).
### (c) Is \(\mathbf{w}\) in the subspace spanned by \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\)? Why?
Examine whether the vector \(\mathbf{w}\) can be written as a linear combination of the vectors \(\mathbf{v}_1, \mathbf{v}_2\), and \(\mathbf{v}_3\). In other words, explore if there exist scalars \(a, b, c\) such that \( a\mathbf{v}_1 + b\mathbf{v}_2 + c\mathbf{v}_3 = \mathbf{w} \). If so,
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