# linear independence of corresponding columns of unitary similar matrices

$$A$$ and $$B$$ are $$n\times n$$ matrices over $$\mathbb{R}$$. Let there exists $$P\in GL_n(\mathbb{R})$$ such that $$P^TP= PP^T=Id$$ (orthogonal) and $$A=PBP^T$$. Here, $$P^T$$ is transpose of $$P$$.

Let $$A=[A_1,A_2,\cdots,A_n]$$ and $$B=[B_1,B_2,\cdots,B_n]$$, where $$A_i,B_j$$ are column vectors of $$A,B$$ respectively.

If $$\{A_1,A_2,\cdots,A_k\}$$ (for $$k\leq n$$) is set of linearly independent columns of A. Can we say $$\{B_1,B_2,\cdots,B_k\}$$ is also set of linearly independent columns of $$B$$?