Let's say I have the following family
$f = ((1,0,1,0),(-1,0,-1,0), (1,1,1,1),(0,1,0,1), (1,2,1,0))$.
I want to first find its rank. I can say, by eliminating all the colinear vectors, that the rank of my family is equal to
rank of $f = \dim(span((1,0,1,0),(0,1,0,1), (1,2,1,0))) = 3$.
Now, I'm very confused about the the whole idea of dimensions. When someone say that the dimension of a basis is always equal to the dimension of the vector space, they're speaking about the number of vectors in the basis, correct? Does this mean that a family of $3$ vectors can be a basis, even if the vectors in the family are of dimension $4$.
In the particular example I gave, can I say that $((1,0,1,0),(0,1,0,1),(1,2,1,0))$ is a basis of $\mathbb R$$^3$? Or does that make no sense? Thank you.