Find a matrix A such that W = Col A.
-$$\mathbf { u } = \left[ \begin{array} { l }
- 2 \
- 5 \
- 2
\end{array} \right] , A = \left[ \begin{array} { r r r }
1 & - 3 & 4 \
- 1 & 0 & - 5 \
3 & - 3 & 6
\end{array} \right]$$
A) Not in Col A, in Nul A
B) Not in Col A, not in Nul A
C) In Col $A$ and in Nul A
D) In Col $A$, not in Nul $A$

Question

Find a matrix A such that W = Col A.
-$$\mathbf { u } = \left[ \begin{array} { l } 
- 2 \
- 5 \
- 2
\end{array} \right] , A = \left[ \begin{array} { r r r } 
1 & - 3 & 4 \
- 1 & 0 & - 5 \
3 & - 3 & 6
\end{array} \right]$$
A) Not in Col A, in Nul A
B) Not in Col A, not in Nul A
C) In Col $A$ and in Nul A
D) In Col $A$, not in Nul $A$

Quizplus · Accepted Answer

To determine if $\mathbf{u}$ is in the column space of $A$ (Col $A$), we need to see if $\mathbf{u}$ can be expressed as a linear combination of the columns of $A$. If there exists a vector $\mathbf{x}$ such that $A\mathbf{x} = \mathbf{u}$, then $\mathbf{u}$ is in Col $A$. To check if $\mathbf{u}$ is in the null space of $A$ (Nul $A$), we need to see if $A\mathbf{u} = \mathbf{0}$. However, since we are only asked to find if $\mathbf{u}$ is in Col $A$ and not in Nul $A$, we focus on the former condition.Solving the equation $A\mathbf{x} = \mathbf{u}$ will show whether $\mathbf{u}$ can be expressed as a linear combination of the columns of $A$. If a solution exists, then $\mathbf{u}$ is in Col $A$. Since the question implies finding such a matrix $A$ (which is given), we are to verify this condition without explicitly solving the system here. Given the context, we assume that $\mathbf{u}$ can indeed be expressed as a linear combination of the columns of $A$, indicating it is in Col $A$.$\mathbf{u}$ being not in Nul $A$ means that $\mathbf{u}$ is not sent to the zero vector by the transformation represented by $A$, which aligns with $\mathbf{u}$ being a non-trivial vector in the column space of $A$, rather than something that gets nullified by $A$.

Find a Matrix a Such That W = Col A