How can it be shown that B is a basis for H? O A. The augmented matrix is upper triangular and row equivalent to [B x] therefore, B forms a basis for H. OB. H is the Span{V₁, V₂, V3} and B= {V₁, V2, V3} so therefore B must form a basis for H. OC. The first three columns of the augmented matrix are pivot columns and therefore B forms a basis for H. O D. The augmented matrix shows that the system of equations is consistent and therefore B forms a basis for H. How can it be shown that x is in H? OA. The first three columns of the augmented matrix are pivot columns and therefore x is in H. OB. The last row of the augmented matrix has zero for all entries and this implies that x must be in H. OC. The augmented matrix is upper triangular and row equivalent to [B x ], therefore x is in H because H is the Span (V₁, V₂, V3} and B= {V₁, V₂, V3}. O D. The augmented matrix shows that the system of equations is consistent and therefore x is in H. The B-coordinate vector of is [x] =
How can it be shown that B is a basis for H? O A. The augmented matrix is upper triangular and row equivalent to [B x] therefore, B forms a basis for H. OB. H is the Span{V₁, V₂, V3} and B= {V₁, V2, V3} so therefore B must form a basis for H. OC. The first three columns of the augmented matrix are pivot columns and therefore B forms a basis for H. O D. The augmented matrix shows that the system of equations is consistent and therefore B forms a basis for H. How can it be shown that x is in H? OA. The first three columns of the augmented matrix are pivot columns and therefore x is in H. OB. The last row of the augmented matrix has zero for all entries and this implies that x must be in H. OC. The augmented matrix is upper triangular and row equivalent to [B x ], therefore x is in H because H is the Span (V₁, V₂, V3} and B= {V₁, V₂, V3}. O D. The augmented matrix shows that the system of equations is consistent and therefore x is in H. The B-coordinate vector of is [x] =
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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![## Linear Algebra: Bases and Coordinate Vectors
### Problem Statement
Given the subspace \( H = \text{Span}\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \) and the set \( B = \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \), show that \( B \) is a basis for \( H \) and that \( \mathbf{x} \) is in \( H \). Also, find the \( B \)-coordinate vector of \( \mathbf{x} \) for the given vectors.
### Given Vectors
\[
\mathbf{v}_1 = \begin{bmatrix} -7 \\ 5 \\ -7 \\ 3 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 9 \\ -2 \\ 8 \\ -5 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} -8 \\ 6 \\ -9 \\ -5 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} 25 \\ 9 \\ 18 \\ -26 \end{bmatrix}
\]
### Matrix Representation
The augmented matrix representing the set \( \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{x} \} \):
\[
\left[ \begin{array}{ccc|c}
-7 & 9 & -8 & 25 \\
5 & -2 & 6 & 9 \\
-7 & 8 & -9 & 18 \\
3 & -5 & -5 & -26 \\
\end{array} \right]
\]
### Questions and Explanatory Options
#### How can it be shown that \( B \) is a basis for \( H \)?
**Option A**: The augmented matrix is upper triangular and row equivalent to \( \left[ B \ \mathbf{x} \right] \), therefore, \( B \) forms a basis for \( H \).
**Option B**: \( H \) is the Span\(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) and \( B = \{\mathbf{v}_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2F8101eeb3-bd00-4f4b-87aa-23dea74eda15%2Fdoncmpi_processed.png&w=3840&q=75)
Transcribed Image Text:## Linear Algebra: Bases and Coordinate Vectors
### Problem Statement
Given the subspace \( H = \text{Span}\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \) and the set \( B = \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \), show that \( B \) is a basis for \( H \) and that \( \mathbf{x} \) is in \( H \). Also, find the \( B \)-coordinate vector of \( \mathbf{x} \) for the given vectors.
### Given Vectors
\[
\mathbf{v}_1 = \begin{bmatrix} -7 \\ 5 \\ -7 \\ 3 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 9 \\ -2 \\ 8 \\ -5 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} -8 \\ 6 \\ -9 \\ -5 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} 25 \\ 9 \\ 18 \\ -26 \end{bmatrix}
\]
### Matrix Representation
The augmented matrix representing the set \( \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{x} \} \):
\[
\left[ \begin{array}{ccc|c}
-7 & 9 & -8 & 25 \\
5 & -2 & 6 & 9 \\
-7 & 8 & -9 & 18 \\
3 & -5 & -5 & -26 \\
\end{array} \right]
\]
### Questions and Explanatory Options
#### How can it be shown that \( B \) is a basis for \( H \)?
**Option A**: The augmented matrix is upper triangular and row equivalent to \( \left[ B \ \mathbf{x} \right] \), therefore, \( B \) forms a basis for \( H \).
**Option B**: \( H \) is the Span\(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) and \( B = \{\mathbf{v}_
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