If $V$ is a $k$-dimensional linear subspace of $R^n$, then $V$ is the kernel of a linear transformation $T : R^n → R^{n−k}.$

Question

If $V$ is a $k$-dimensional linear subspace of $R^n$, then $V$ is the kernel of a linear transformation $T : R^n → R^{n−k}.$

Basically just looking for whether this is True or False.

Answer 1

Consider the quotient map onto $\Bbb R^n/V\cong\Bbb R^{n-k}$. (Halmos is a good reference on this.)