Change of basis and dimension of subspaces I'm trying to wrap my head around some basic concepts here. I have some vector subspace $\mathcal{V}$ and $\mathcal{W} = \{Ax \mid x \in \mathcal{V}\}$, where $A$ is an orthogonal matrix. Is it always true that $\dim(\mathcal{W}) = \dim(\mathcal{V})$? If $\{v_1, v_2, \dots, v_k\}$ is a basis of $\mathcal{V}$, how can we find a basis of $\mathcal{W}$?

It's enough to assume that $A$ is invertible, which an orthogonal matrix is, to conclude $W=A(V)$ has the same dimension as $V$.
It's because both $A$ and $A^{-1}$ takes any linearly independent set of vectors to a linearly independent set.

 Suppose $v_1,\dots,v_k$ are independent and $\sum_i\lambda_i\,Av_i=0$, then apply $A^{-1}$ to obtain $\sum_i\lambda_iv_i=0$ so each $\lambda_i=0$.

