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 $\mathcal{W}$?

I will assume you are working with finite-dimensional subspaces.

*

*In general, if $W$ is a subspace of $V$ and $\dim (W) = \dim (V)$, then $W = V$, so any basis of $V$ is a basis of $W$.

*You have defined $W$ as the column space of $A$ (span of the columns of $A$). If the columns of $A$ are linearly independent, then the dimension of $W$ is the number of columns of $A$. In your specific situation, $A$ is a square invertible matrix (because it is orthogonal) with $\dim(V)$ columns so $\dim(W) = \dim(V)$.

