I have been under the impression that the dimension of a matrix is simply whatever dimension it lives in. More precisely, if a vector space contained the vectors $(v_1, v_2,...,v_n)$, where each vector contained $3$ components $(a,b,c)$ (for some $a$, $b$ and $c$), then its dimension would be $\Bbb R^3$.
Recently I was told this is not true, and the dimension of this vector space would be $\Bbb R^n$.
In other words, I was under the belief that the dimension is the number of elements that compose the vectors in our vector space, but the dimension is how many vectors the vector space contains?!
Where am I going wrong?