Example 7: Find (a) the rank and nullity of A and (b) a subset of the column vectors of A that forms a basis for the column space of A. -2 -4 4 6. -6 -4 -2 -4 4 9.
Example 7: Find (a) the rank and nullity of A and (b) a subset of the column vectors of A that forms a basis for the column space of A. -2 -4 4 6. -6 -4 -2 -4 4 9.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Example 7:** Find (a) the rank and nullity of \( A \) and (b) a subset of the column vectors of \( A \) that forms a basis for the column space of \( A \).
\[
A = \begin{bmatrix}
-2 & -4 & 4 & 5 \\
3 & 6 & -6 & -4 \\
-2 & -4 & 4 & 9
\end{bmatrix}
\]
**Explanation for Educational Context:**
In this example, matrix \( A \) is a \( 3 \times 4 \) matrix. The problem is asking to determine:
a) The **rank** of \( A \), which is the dimension of the column space of \( A \) (the number of linearly independent columns), and the **nullity** of \( A \), which is the dimension of the null space of \( A \) (the number of linearly independent solutions to the homogeneous equation \( A\mathbf{x} = 0 \)).
b) A subset of the columns of \( A \) that forms a basis for the column space of \( A \), indicating which columns are linearly independent and span the same space as all columns of \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff57fbb07-e1bd-418f-9d01-1252f5b1cb71%2Ff8dfac4a-d1b8-449a-90ee-718153c0f9f7%2Fkh4w41f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example 7:** Find (a) the rank and nullity of \( A \) and (b) a subset of the column vectors of \( A \) that forms a basis for the column space of \( A \).
\[
A = \begin{bmatrix}
-2 & -4 & 4 & 5 \\
3 & 6 & -6 & -4 \\
-2 & -4 & 4 & 9
\end{bmatrix}
\]
**Explanation for Educational Context:**
In this example, matrix \( A \) is a \( 3 \times 4 \) matrix. The problem is asking to determine:
a) The **rank** of \( A \), which is the dimension of the column space of \( A \) (the number of linearly independent columns), and the **nullity** of \( A \), which is the dimension of the null space of \( A \) (the number of linearly independent solutions to the homogeneous equation \( A\mathbf{x} = 0 \)).
b) A subset of the columns of \( A \) that forms a basis for the column space of \( A \), indicating which columns are linearly independent and span the same space as all columns of \( A \).
Expert Solution
Step 1
The given matrix is:
To find rank and nullity of matrix, first we will find the ow reduced echelon form of matrix.
Now applying :
now applying
Step by step
Solved in 4 steps
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