Let $f_1,f_2$ be two orthogonal non zero vectors that span a plane $\pi$. The projection of a vector $w$ onto the plane $\pi$ is given by $Proj_{\pi}(w)= $$\frac{w\cdot n}{f_1\cdot f_1}\cdot f_1 + \frac{w\cdot n}{f_2\cdot f_2}\cdot f_2$. Prove that the function $T : R^3 \to R^3$ defined by $T(w) = Proj_{\pi}(w) $ is a linear transformation.
Things that I know:
A transformation $T : R^m →R^m$ is called a linear transformation if and only if
1)$T(u + v) = T(u) + T(v)$ for all $u$ and $v$ in $R^n$.
2) $T(cv) = cT(v)$ for all $ v$ in $R^n$ and all scalars $c$
So I think I need to prove these two things, but I am not sure how to proceed with that.
Edit for the second part of the proof
For Scaler Multiplication. Proof: for another vector $w ∈ R^n$ and a scalar c, it
$proj_v(u+w) = proj_v(u)+proj_v(w) and proj_v(cu) = c(proj_v(u)$
Best Answer
Part the First (Additivity):
Let $u,v \in \Bbb R^3$. Then
$$\begin{align}\operatorname{Proj}_\pi (u+v) &= \frac{(u+v)\cdot f_1}{f_1\cdot f_1} f_1 + \frac{(u+v)\cdot f_2}{f_2\cdot f_2} f_2 \\ &= \frac{u\cdot f_1+v\cdot f_1}{f_1\cdot f_1} f_1 + \frac{u\cdot f_2+v\cdot f_2}{f_2\cdot f_2} f_2 \\ &= \left[\frac{u\cdot f_1}{f_1\cdot f_1} + \frac{v\cdot f_1}{f_1\cdot f_1}\right] f_1 + \left[\frac{u\cdot f_2}{f_2\cdot f_2} + \frac{v\cdot f_2}{f_2\cdot f_2}\right] f_2 \\ &= \frac{u\cdot f_1}{f_1\cdot f_1} f_1 + \frac{v\cdot f_1}{f_1\cdot f_1} f_1 + \frac{u\cdot f_2}{f_2\cdot f_2} f_2 + \frac{v\cdot f_2}{f_2\cdot f_2} f_2 \\ &= \left[\frac{u\cdot f_1}{f_1\cdot f_1} f_1 + \frac{u\cdot f_2}{f_2\cdot f_2} f_2\right] + \left[\frac{v\cdot f_1}{f_1\cdot f_1} f_1 + \frac{v\cdot f_2}{f_2\cdot f_2} f_2\right] \\ &= \operatorname{Proj}_\pi (u) + \operatorname{Proj}_\pi (v)\end{align}$$
Thus $\operatorname{Proj}_\pi$ is additive.
Part the Second (Homogeneity):
Try this on your own using the above as a guide.