6. Find a basis for each of the four subspaces associated with the matrix: [1 2 3 4] 1 4 6 2 2 2 A = 2 1

Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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**Problem Statement:**

6. Find a basis for each of the four subspaces associated with the matrix:

\[ 
A = \begin{bmatrix} 
1 & 2 & 3 & 4 \\ 
2 & 1 & 4 & 6 \\ 
1 & 2 & 2 & 2 
\end{bmatrix} 
\]

**Explanation:**

You need to find a basis for the following four fundamental subspaces related to the matrix \( A \):

1. **Column Space (C(A))**: The set of all linear combinations of the columns of \( A \).
2. **Row Space (C(A\(^T\)))**: The set of all linear combinations of the rows of \( A \), which is the same as the column space of the transpose of \( A \).
3. **Null Space (N(A))**: The set of solutions to the homogeneous equation \( A\mathbf{x} = \mathbf{0} \).
4. **Left Null Space (N(A\(^T\)))**: The null space of the transpose of \( A \), associated with the equation \( A^T\mathbf{y} = \mathbf{0} \).

To find these bases, use techniques like row reduction to echelon form, solving systems of linear equations, and employing the Gram-Schmidt process if necessary.
Transcribed Image Text:**Problem Statement:** 6. Find a basis for each of the four subspaces associated with the matrix: \[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 6 \\ 1 & 2 & 2 & 2 \end{bmatrix} \] **Explanation:** You need to find a basis for the following four fundamental subspaces related to the matrix \( A \): 1. **Column Space (C(A))**: The set of all linear combinations of the columns of \( A \). 2. **Row Space (C(A\(^T\)))**: The set of all linear combinations of the rows of \( A \), which is the same as the column space of the transpose of \( A \). 3. **Null Space (N(A))**: The set of solutions to the homogeneous equation \( A\mathbf{x} = \mathbf{0} \). 4. **Left Null Space (N(A\(^T\)))**: The null space of the transpose of \( A \), associated with the equation \( A^T\mathbf{y} = \mathbf{0} \). To find these bases, use techniques like row reduction to echelon form, solving systems of linear equations, and employing the Gram-Schmidt process if necessary.
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