the null space, and the row space of the matrix 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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![**Find Bases for the Null Space and Row Space of a Matrix**
Given the matrix \( A \), we need to find the bases for its null space and row space. The matrix \( A \) is:
\[
A = \begin{bmatrix}
1 & 3 & 4 & 6 & 8 \\
2 & -10 & -8 & -4 & -16 \\
-1 & 0 & -1 & 0 & -2 \\
3 & 4 & 7 & 13 & 14
\end{bmatrix}
\]
**Null Space of A**
The basis vectors for the null space of \( A \) are:
\[
\mathbf{v_1} = \begin{bmatrix}
-1 \\
-1 \\
1 \\
0 \\
0
\end{bmatrix}
\]
\[
\mathbf{v_2} = \begin{bmatrix}
-2 \\
-2 \\
0 \\
0 \\
1
\end{bmatrix}
\]
**Row Space of A**
The basis vectors for the row space of \( A \) are:
\[
\mathbf{r_1} = \begin{bmatrix}
1 \\
3 \\
4 \\
6 \\
8
\end{bmatrix}
\]
\[
\mathbf{r_2} = \begin{bmatrix}
0 \\
1 \\
1 \\
1 \\
2
\end{bmatrix}
\]
\[
\mathbf{r_3} = \begin{bmatrix}
0 \\
0 \\
0 \\
1 \\
0
\end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9b53ebc-804c-49f3-b917-c4e1c4975b52%2F63cd2aae-201c-46f0-adeb-66c923bac75b%2Fsal9mp_processed.png&w=3840&q=75)
Transcribed Image Text:**Find Bases for the Null Space and Row Space of a Matrix**
Given the matrix \( A \), we need to find the bases for its null space and row space. The matrix \( A \) is:
\[
A = \begin{bmatrix}
1 & 3 & 4 & 6 & 8 \\
2 & -10 & -8 & -4 & -16 \\
-1 & 0 & -1 & 0 & -2 \\
3 & 4 & 7 & 13 & 14
\end{bmatrix}
\]
**Null Space of A**
The basis vectors for the null space of \( A \) are:
\[
\mathbf{v_1} = \begin{bmatrix}
-1 \\
-1 \\
1 \\
0 \\
0
\end{bmatrix}
\]
\[
\mathbf{v_2} = \begin{bmatrix}
-2 \\
-2 \\
0 \\
0 \\
1
\end{bmatrix}
\]
**Row Space of A**
The basis vectors for the row space of \( A \) are:
\[
\mathbf{r_1} = \begin{bmatrix}
1 \\
3 \\
4 \\
6 \\
8
\end{bmatrix}
\]
\[
\mathbf{r_2} = \begin{bmatrix}
0 \\
1 \\
1 \\
1 \\
2
\end{bmatrix}
\]
\[
\mathbf{r_3} = \begin{bmatrix}
0 \\
0 \\
0 \\
1 \\
0
\end{bmatrix}
\]
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