Are $V=<(1,0,1),(1,1,1)>$ and $W=<(1,0,1),(0,1,0)>$ equivalent?

Question

I just wonder if we can treat vector spaces as matrices?

For example, are subspaces

$V=<(1,0,1),(1,1,1)>$ and $W=<(1,0,1),(0,1,0)>$ analogous?

Answer 1

For example, are subspaces

$\color{blue}{V}=\{(1,0,1),(1,1,1)\}$ and $W=\{(1,0,1),(0,1,0)\}$ analogous?

It's easier to use different names so I renamed the first one to $V$.

Note that in the usual notation, you didn't give subspaces but simply two sets containing two elements (each).

The sets are (clearly) different ($V \ne W$) but they do span the same subspace: $$\mbox{span}\,V=\mbox{span}\,W$$ In your case, this is easy to see because $(1,0,1)+(0,1,0)=(1,1,1)$.

Answer 2

Yes the two subspaces are equivalent or equal indeed

$$(1,1,1)=(1,0,1)+(0,1,0)$$

and then

$$Span\{(1,0,1),(1,1,1)\}=Span\{(1,0,1),(0,1,0)\}$$