How do I prove that a projection onto a subspace is a linear transformation ?
Given that V is a vector space, and M is a subspace of V.
I know these two facts:
i) There exists a subspace $N$ such that every vector $v \in V$ can be written uniquely as $v = x + y$ for some $x \in M$ and $y \in N$.
ii) $P$ is given by $P(x + y) = x$, for all $x \in M$ and $y \in N$
Best Answer
If $v=x+y$ and $w=x'+y'$, with $x,x'\in M$ and $y,y'\in N$, then\begin{align}P(v+w)&=P(x+y+x'+y')\\&=P(x+x'+y+y')\\&=x+x'\\&=P(v)+P(w).\end{align}Can you prove now that if $\lambda$ is a scalar, then $P(\lambda v)=\lambda P(v)$?