Solve this true/flase linear algebra question. Please show your work.
Transcribed Image Text:**Null Space of a Matrix**
Given the following matrix \( A \) and vector \( \mathbf{w} \):
\[ A = \begin{bmatrix} -8 & -2 & -9 \\ 6 & 4 & 8 \\ 4 & 0 & 4 \end{bmatrix} \quad \text{and} \quad \mathbf{w} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} \]
Is it true that \( \mathbf{w} \) is in the Null Space of \( A \) (Null(A))?
**Options:**
- True
- False
**Explanation:**
To determine if \( \mathbf{w} \) is in the Null Space of \( A \), we need to verify if the matrix-vector product \( A\mathbf{w} \) results in the zero vector. In other words, we need to check if:
\[ A\mathbf{w} = \mathbf{0} \]
Perform the matrix multiplication \( A\mathbf{w} \) to find out. If the result is the zero vector, then \( \mathbf{w} \) is in the Null Space of \( A \); otherwise, it is not.
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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![**Null Space of a Matrix**
Given the following matrix \( A \) and vector \( \mathbf{w} \):
\[ A = \begin{bmatrix} -8 & -2 & -9 \\ 6 & 4 & 8 \\ 4 & 0 & 4 \end{bmatrix} \quad \text{and} \quad \mathbf{w} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} \]
Is it true that \( \mathbf{w} \) is in the Null Space of \( A \) (Null(A))?
**Options:**
- True
- False
**Explanation:**
To determine if \( \mathbf{w} \) is in the Null Space of \( A \), we need to verify if the matrix-vector product \( A\mathbf{w} \) results in the zero vector. In other words, we need to check if:
\[ A\mathbf{w} = \mathbf{0} \]
Perform the matrix multiplication \( A\mathbf{w} \) to find out. If the result is the zero vector, then \( \mathbf{w} \) is in the Null Space of \( A \); otherwise, it is not.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2e16516-2d4e-4d7b-b0b6-ee6bc0c14cd9%2Fece55ccf-89c8-4cae-aca0-9af681866b5a%2F4embbin_processed.jpeg&w=3840&q=75)
