Find bases for the column space, the row space, and the null space of matrix A. You should verify that the Rank-Nullity Theorem holds. An equivalent echelon form of matrix A is given to make your work easier. 1 0 2 0 1 3 0 0 0 1 2 8 A = 3 3 15 ~ 3 8 30 Basis for the column space of A = { Basis for the row space of A = { Basis for the null space of A={| | || }
Find bases for the column space, the row space, and the null space of matrix A. You should verify that the Rank-Nullity Theorem holds. An equivalent echelon form of matrix A is given to make your work easier. 1 0 2 0 1 3 0 0 0 1 2 8 A = 3 3 15 ~ 3 8 30 Basis for the column space of A = { Basis for the row space of A = { Basis for the null space 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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![**Task**: Find bases for the column space, the row space, and the null space of matrix A. You should verify that the Rank-Nullity Theorem holds. An equivalent echelon form of matrix A is given to make your work easier.
Given Matrix:
\[
A = \begin{bmatrix}
1 & 2 & 8 \\
3 & 3 & 15 \\
3 & 8 & 30
\end{bmatrix}
\sim
\begin{bmatrix}
1 & 0 & 2 \\
0 & 1 & 3 \\
0 & 0 & 0
\end{bmatrix}
\]
**Basis for the Column Space of A**:
\[
\left\{ \begin{bmatrix} \\ \\ \end{bmatrix}, \begin{bmatrix} \\ \\ \end{bmatrix} \right\}
\]
**Basis for the Row Space of A**:
\[
\left\{ \begin{bmatrix} \\ \\ \end{bmatrix}, \begin{bmatrix} \\ \\ \end{bmatrix} \right\}
\]
**Basis for the Null Space of A**:
\[
\left\{ \begin{bmatrix} \\ \\ \end{bmatrix} \right\}
\]
The matrix A is first reduced to its row-echelon form to identify the pivotal columns, which help determine the bases for the column and row spaces. The zero row indicates a reduction in rank.
The Rank-Nullity Theorem can be verified by ensuring that the sum of the rank and the nullity (dimension of the null space) equals the number of columns in the matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf673b33-84d3-4207-a3d8-77b439e8ab65%2F70845e3f-b2c1-48e5-9291-0c2d22399528%2Fmkfcqac_processed.png&w=3840&q=75)
Transcribed Image Text:**Task**: Find bases for the column space, the row space, and the null space of matrix A. You should verify that the Rank-Nullity Theorem holds. An equivalent echelon form of matrix A is given to make your work easier.
Given Matrix:
\[
A = \begin{bmatrix}
1 & 2 & 8 \\
3 & 3 & 15 \\
3 & 8 & 30
\end{bmatrix}
\sim
\begin{bmatrix}
1 & 0 & 2 \\
0 & 1 & 3 \\
0 & 0 & 0
\end{bmatrix}
\]
**Basis for the Column Space of A**:
\[
\left\{ \begin{bmatrix} \\ \\ \end{bmatrix}, \begin{bmatrix} \\ \\ \end{bmatrix} \right\}
\]
**Basis for the Row Space of A**:
\[
\left\{ \begin{bmatrix} \\ \\ \end{bmatrix}, \begin{bmatrix} \\ \\ \end{bmatrix} \right\}
\]
**Basis for the Null Space of A**:
\[
\left\{ \begin{bmatrix} \\ \\ \end{bmatrix} \right\}
\]
The matrix A is first reduced to its row-echelon form to identify the pivotal columns, which help determine the bases for the column and row spaces. The zero row indicates a reduction in rank.
The Rank-Nullity Theorem can be verified by ensuring that the sum of the rank and the nullity (dimension of the null space) equals the number of columns in the matrix.
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