2 6 8 4 Let u - 4 and A = 0 2 -2 . Is u in the subset of R° spanned by the columns of A? Why or why not? 1 2 0 Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal for each matrix element.) O A. No, the reduced row echelon form of the augmented matrix is , which is an inconsistent system. O B. Yes, multiplying A by the vector writes u as a linear combination of the columns of A.
2 6 8 4 Let u - 4 and A = 0 2 -2 . Is u in the subset of R° spanned by the columns of A? Why or why not? 1 2 0 Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal for each matrix element.) O A. No, the reduced row echelon form of the augmented matrix is , which is an inconsistent system. O B. Yes, multiplying A by the vector writes u as a linear combination of the columns of A.
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
![**Matrix Membership in Subspace**
In this exercise, we are exploring whether the vector \( \mathbf{u} \) is within the subset of \(\mathbb{R}^3\) spanned by the columns of matrix \( \mathbf{A} \).
Given:
\[
\mathbf{u} = \begin{bmatrix}
2 \\
-4 \\
1 \\
\end{bmatrix}
\]
and
\[
\mathbf{A} = \begin{bmatrix}
6 & 8 & 4 \\
0 & 2 & -2 \\
1 & 2 & 0 \\
\end{bmatrix}.
\]
**Question:** Is \( \mathbf{u} \) in the subset of \(\mathbb{R}^3\) spanned by the columns of \( \mathbf{A} \)? Why or why not?
**To Solve:**
Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal for each matrix element.)
**Options:**
- **A.** No, the reduced row echelon form of the augmented matrix is \(\boxed{\quad}\), which is an inconsistent system.
- **B.** Yes, multiplying \( \mathbf{A} \) by the vector \(\boxed{\quad}\) writes \( \mathbf{u} \) as a linear combination of the columns of \( \mathbf{A} \).
To decide which option is correct, we would proceed to set up the augmented matrix and perform row reduction to see if the system is consistent or not.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2F0c9fc55c-cb99-424a-b8d3-1f411f17dcee%2F57l374l_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Membership in Subspace**
In this exercise, we are exploring whether the vector \( \mathbf{u} \) is within the subset of \(\mathbb{R}^3\) spanned by the columns of matrix \( \mathbf{A} \).
Given:
\[
\mathbf{u} = \begin{bmatrix}
2 \\
-4 \\
1 \\
\end{bmatrix}
\]
and
\[
\mathbf{A} = \begin{bmatrix}
6 & 8 & 4 \\
0 & 2 & -2 \\
1 & 2 & 0 \\
\end{bmatrix}.
\]
**Question:** Is \( \mathbf{u} \) in the subset of \(\mathbb{R}^3\) spanned by the columns of \( \mathbf{A} \)? Why or why not?
**To Solve:**
Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal for each matrix element.)
**Options:**
- **A.** No, the reduced row echelon form of the augmented matrix is \(\boxed{\quad}\), which is an inconsistent system.
- **B.** Yes, multiplying \( \mathbf{A} \) by the vector \(\boxed{\quad}\) writes \( \mathbf{u} \) as a linear combination of the columns of \( \mathbf{A} \).
To decide which option is correct, we would proceed to set up the augmented matrix and perform row reduction to see if the system is consistent or not.
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