Determine whether b is in the column space of A.
-$$A = \left[ \begin{array} { r r r }
1 & 2 & - 3 \
1 & 4 & - 6 \
- 3 & - 2 & 5
\end{array} \right] , \mathbf { b } = \left[ \begin{array} { r }
1 \
- 2 \
- 3
\end{array} \right]$$
A) No
B) Yes

Question

Determine whether b is in the column space of A.
-$$A = \left[ \begin{array} { r r r } 
1 & 2 & - 3 \
1 & 4 & - 6 \
- 3 & - 2 & 5
\end{array} \right] , \mathbf { b } = \left[ \begin{array} { r } 
1 \
- 2 \
- 3
\end{array} \right]$$
A) No
B) Yes

Quizplus · Accepted Answer

To determine if $\mathbf{b}$ is in the column space of $A$, we need to check if there exists a vector $\mathbf{x}$ such that $A\mathbf{x} = \mathbf{b}$. This can be done by augmenting matrix $A$ with vector $\mathbf{b}$ and reducing it to row echelon form or by checking the rank of $A$ and the augmented matrix $A|\mathbf{b}$. If the system is consistent (has a solution), then $\mathbf{b}$ is in the column space of $A$. Given the matrices, if we perform these operations, we would find that the system is indeed consistent, indicating that $\mathbf{b}$ is in the column space of $A$.

Determine Whether B Is in the Column Space of A