Can span contain redundant vectors in its set

linear algebra

I suppose it is probably answered somewhere, but I cannot for the life of me find the post. So here goes:

If $\{v_1,v_2,v_3\}$ spans $\mathbb{R}^3$, then could $\{v_1,v_2,v_3,v_4\}$ also span $\mathbb{R}^3$, where $v_4\ne v_1,v_2,v_3$?

Best Answer

Yes. In fact, we have the more general result:

Theorem. Suppose that $S$ and $T$ are subsets of a vector space $V$ with $S\subseteq T$. Then $\operatorname{Span}(S)\subseteq\operatorname{Span}(T)$.

In our case, $S=\{v_1, v_2, v_3\}$ and $T=\{v_1, v_2, v_3, v_4\}$ where each $v_i$ is a vector in $\Bbb R^3$. Then $\operatorname{Span}(S)=\Bbb R^3$ along with our above theorem gives $$ \Bbb R^3=\operatorname{Span}(S)\subseteq\operatorname{Span}(T)\subseteq\Bbb R^3 $$ It follows that $\operatorname{Span}(T)=\Bbb R^3$.

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