Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset? Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does?  Since it's a subset? A little unclear about this...

In general: if $A$ spans $V$ and $A\subset B$, then $B$ spans $V$. The converse is obviously false, but if $B$ spans $V$ and $b\in B$ is a linear combination of elements of $B\backslash\{b\}$, then $B\backslash\{b\}$ spans $V$.
