Suppose $U_1$ and $U_2$ are subspaces of a finite-dimensional vector space.
Let $u_1,...,u_m$ be a basis of $U_1\cap U_2$, thus dimension of the intersection is $m$.
$\textbf{The part I don't understand is:}$
Because $u_1,...,u_m$ is a basis of $U_1\cap U_2$, it is linearly independent in $U_1$.
Why is this true?
Reference:
Axler, Sheldon J. $\textit{Linear Algebra Done Right}$, New York: Springer, 2015.