How Span relates to Consistency of Matrix-Vector Product

• We know $A\vec{c}=\vec{x}$ basically means that $\vec{x}$ is a linear combination of the entries of $\vec{x}$ and columns of $A$.

• Since $A=[{\vec{v}}_1,...,{\vec{v}}_k]$, $\vec{x}$ is basically a linear combination of ${\vec{v}}_i$ and elements of $\vec{c}$ (from (4) in Matrix-Vector Product).

• For $\vec{x}\in \text{Span }B$, $\vec{x}$ needs to be a linear combination of the vectors in $B$.

• Since $B=\{{\vec{v}}_1,...,{\vec{v}}_k\}$ and $A\vec{c}=\vec{x}$ (hence $\vec{x}$ is a linear combination of ${\vec{v}}_i$), $\vec{x}\in \text{Span }B$.

• This enables us to check if $\vec{x}\in \text{Span }B$ by just checking if $A\vec{c}=\vec{x}$ is consistent.

• How to check consistency of matrix-vector product by carrying it to REF