How do I prove that 2 vector spaces are equal if they have different number of vectors that span them?

Question

For example, in 4-dimensions, vector space V is spanned by the linearly independent vectors x, y, and z and vector space U is spanned by the linearly independent vectors r, and s. Would I try to prove that the vectors x, y, and z can span U and vectors r, and s can span V?

Answer 1

If $S$ and $S'$ are subsets of a vector space $V$, then $\operatorname{span}(S)=\operatorname{span}(S')$ if and only if each vector of $S$ is a linear combination of elements of $S'$ and vice-versa.

Consider, for instance, the case in which:

• $V=\mathbb{R}^3$;
• $S=\bigl\{(1,-1,0),(1,0,-1),(0,1,-1)\bigr\}$;
• $S'=\bigl\{(2,-1,-1),(-1,-1,2)\bigr\}$.

I claim that $\operatorname{span}(S)=\operatorname{span}(S')$. In order to prove it, all I have to do is to note that:

• $\displaystyle(1,-1,0)=\frac23(2,-1,-1)+\frac13(-1,-1,2)$;
• $\displaystyle(1,0,-1)=\frac13(2,-1,-1)-\frac13(-1,-1,2)$;
• $\displaystyle(0,1,-1)=-\frac13(2,-1,-1)-\frac23(-1,-1,2)$;
• $\displaystyle(2,-1,-1)=(1,-1,0)+(1,0,-1)$;
• $\displaystyle(-1,-1,2)=-(1,0,-1)-(0,1,-1)$.