Suppose that we have $p$ linearly independent vectors $x_1,\ldots,x_p\in\mathbb R^n$ such that each of these vectors can be expressed as a linear combination of orthogonal vectors $z_1,\ldots,z_p\in\mathbb R$. I am trying to show that the subspaces spanned by $x_1,\ldots,x_p$ and $z_1,\ldots,z_p$ are the same.
$z_1,\ldots,z_p$ span the subspace spanned by the vectors $x_1,\ldots,x_p\in\mathbb R^n$ because $x_1,\ldots,x_p$ can be expressed as linear combinations of $z_1,\ldots,z_p$. What could be the argument we could use to make sure that $z_1,\ldots,z_p$ cannot span a larger subspace? It seems that a proof by contradiction is the way to go but I am not sure how to proceed.
This is question is related to the Gram-Schmidt process.
Any help is much appreciated!