I want to show:
Given $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times p}$ with $rank(B)=p$ and $C(B)\subset C(A)$. Prove $B'AB$ is positive definite, where $C(B)$ denotes the column space of matrix $B$.
The steps below are what I have done so far.
Since $B'AB \in \mathbb{R}^{p \times p}$ , intuitively, I consider to prove its definiteness from the definition; i.e., for arbitrary nonzero (column) vector $x\in\mathbb R^p$, it suffices to show that $x'B'ABx >0$.
By $rank(B)=p$ we have $B$ is column full rank, then it follows $B'B$ is positive definite ($x'B'Bx >0$). Then how to use the condition $C(B)\subset C(A)$ to get $x'B'ABx >0$?