Column space of matrix X

Question

How to show that C(X)=C(XA) when A is a p by p nonsingular matrix?

X is a n by p full rank matrix and we can decompose it into (X1,X2)

Answer 1

The elements of $C(X)$ are of the form $Xv$ for some vector $v \in \mathbb{R}^p$; the elements of $C(XA)$ are of the form $XAw$ for some $w \in \mathbb{R}^p$.

To prove $C(X) \supseteq C(XA)$, simply note that if $XAw in C(XA)$, then letting $v = Aw$ yields $XAw=Xv \in C(X)$.

Conversely, to show $C(X) \subseteq C(XA)$, note that if $Xv \in C(X)$ then letting $w = A^{-1} v$ yields $Xv = XAw \in C(XA)$.