where B A = = H (a) Find At -6 rref (A). -1 -2 TATT -1 7 -4 -18 10 9 NAAN -1 1 -3 2 17 (e) Find a basis for the Null(A¹). 6 -26 17 12 (b) Find a basis for the Null(A) = Null(B) (c) Find a basis for the row(A) = row(B) B = (d) Verify that each basis vector of Null(A) is orthogonal to every basis vector for row (A) (f) Find a basis for the row (A) = column(A) [102 1 0 3 0 1 4 -6 0 -1 000 0 1 2 000 0 0 0 (g) Verify that each basis vector of Null(A) is orthogonal to every basis vector for column (A)
where B A = = H (a) Find At -6 rref (A). -1 -2 TATT -1 7 -4 -18 10 9 NAAN -1 1 -3 2 17 (e) Find a basis for the Null(A¹). 6 -26 17 12 (b) Find a basis for the Null(A) = Null(B) (c) Find a basis for the row(A) = row(B) B = (d) Verify that each basis vector of Null(A) is orthogonal to every basis vector for row (A) (f) Find a basis for the row (A) = column(A) [102 1 0 3 0 1 4 -6 0 -1 000 0 1 2 000 0 0 0 (g) Verify that each basis vector of Null(A) is orthogonal to every basis vector for column (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
please naswer this and make sure the writing is understandable.
![Sure, here is the transcription with explanations:
---
Consider the matrices:
\[ A = \begin{bmatrix}
1 & -1 & -2 & 7 & 1 & 6 \\
-6 & 2 & -4 & -18 & -3 & -26 \\
4 & -1 & 4 & 10 & 2 & 17 \\
3 & -1 & 2 & 9 & 1 & 12
\end{bmatrix}, \quad
B = \begin{bmatrix}
1 & 0 & 2 & 1 & 0 & 3 \\
0 & 1 & 4 & -6 & 0 & -1 \\
0 & 0 & 0 & 0 & 1 & 2 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} \]
where \( B = \text{rref}(A) \).
### Tasks
(a) **Find \( A^t \)**
(b) **Find a basis for the \(\text{Null}(A) = \text{Null}(B)\)**
(c) **Find a basis for the \(\text{row}(A) = \text{row}(B)\)**
(d) **Verify that each basis vector of \(\text{Null}(A)\) is orthogonal to every basis vector for \(\text{row}(A)\)**
(e) **Find a basis for the \(\text{Null}(A^t)\)**
(f) **Find a basis for the \(\text{row}(A^t) \cong \text{column}(A)\)**
(g) **Verify that each basis vector of \(\text{Null}(A^t)\) is orthogonal to every basis vector for \(\text{column}(A)\)**
---
### Explanation
- Matrix \( A \) is a \( 4 \times 6 \) matrix, and matrix \( B \) is the row-reduced echelon form (rref) of \( A \).
- Task (a) involves finding the transpose of matrix \( A \).
- Tasks (b) and (e) focus on finding bases for the null spaces of matrices.
- Tasks (c) and (f) involve finding bases for the row space and proving its equivalence to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2903bb2a-771d-40b7-a62d-703de2e08baa%2F0f2641a6-0089-45de-a32e-01a704102d92%2Fys1j93b_processed.png&w=3840&q=75)
Transcribed Image Text:Sure, here is the transcription with explanations:
---
Consider the matrices:
\[ A = \begin{bmatrix}
1 & -1 & -2 & 7 & 1 & 6 \\
-6 & 2 & -4 & -18 & -3 & -26 \\
4 & -1 & 4 & 10 & 2 & 17 \\
3 & -1 & 2 & 9 & 1 & 12
\end{bmatrix}, \quad
B = \begin{bmatrix}
1 & 0 & 2 & 1 & 0 & 3 \\
0 & 1 & 4 & -6 & 0 & -1 \\
0 & 0 & 0 & 0 & 1 & 2 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} \]
where \( B = \text{rref}(A) \).
### Tasks
(a) **Find \( A^t \)**
(b) **Find a basis for the \(\text{Null}(A) = \text{Null}(B)\)**
(c) **Find a basis for the \(\text{row}(A) = \text{row}(B)\)**
(d) **Verify that each basis vector of \(\text{Null}(A)\) is orthogonal to every basis vector for \(\text{row}(A)\)**
(e) **Find a basis for the \(\text{Null}(A^t)\)**
(f) **Find a basis for the \(\text{row}(A^t) \cong \text{column}(A)\)**
(g) **Verify that each basis vector of \(\text{Null}(A^t)\) is orthogonal to every basis vector for \(\text{column}(A)\)**
---
### Explanation
- Matrix \( A \) is a \( 4 \times 6 \) matrix, and matrix \( B \) is the row-reduced echelon form (rref) of \( A \).
- Task (a) involves finding the transpose of matrix \( A \).
- Tasks (b) and (e) focus on finding bases for the null spaces of matrices.
- Tasks (c) and (f) involve finding bases for the row space and proving its equivalence to
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