Linear Dependence of Vectors and Matrices Let $W=\left \{ w_{1},...,w_{m} \right \}$ and $V=\left \{ v_{1},...,v_{n} \right \}$ for $m>n$ where W and V are both independent sets of vectors. ($W$ being an independent set of vectors means $w_1,\cdots,w_m$ are independent. Similarly for $V$)
Why the following cannot be true?

$$\begin{bmatrix}
w_{1}\\ 
w_{2}\\
\vdots \\
w_{m} \end{bmatrix}\neq\begin{bmatrix}
a_{11} & &a_{1n} \\ 
\vdots &\ddots &\vdots\\ 
 a_{m1}&  & a_{mn}\\
\end{bmatrix}
\begin{bmatrix}
v_{1}\\ 
v_{2}\\
\vdots \\
v_{n} \end{bmatrix}$$


Suppose otherwise one has the equality. Then one can write that 
$$
B=AC
$$
where $rank(B)=m$ and $rank(C)=n$. Now, by a property of matrix ranks:
$$
m=rank(B)\leq\min\{rank(A),rank(C)\}=\min\{rank(A),n\}
$$
which is impossible because by the assumption $m>n$.  

[Added later.] I like Martin's answer better than mine: if the equality is true, then it follows that $span(W)\subset span(V)$, which is impossible since $m>n$.
