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$A$ 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 $B$?

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