Question
Let $F : \Re^3 → \Re^3$ be a linear transformation with $F (1, 0, 0) = (1, 1, −1)$, $F(1,1,0) = (2,−1,0)$ and $F(1,1,1) = (3,0,−1)$.
Let $G : \Re^3 → \Re^3$ be orthogonal projection onto the line through $(1, 2, a)$ for $a ∈ \Re$.
Find the value of a for which $G◦F = 0$.
Note that $G◦F = 0$ means that $G ◦ F (x, y, z) = 0$ for all $(x, y, z) ∈ \Re^3$.
My Solution
Okay the main problem I have is that there is too much information here, that it is confusing and I don't know how to deal with it.
Anyway this is what I know:
It is clear that $(1, 1, −1),(2,−1,0),(3,0,−1)$ forms an orthonormal basis because $(1,0,0),(1,1,0),(1,1,1)$ is linearly independent.
So, when they say that $G$ to be the orthogonal projection through $(1,2,a)$ It means that:
$$w_1=(u\cdot v_1)v_1+(u\cdot v_2)v_2+(u\cdot v_3)v_3$$
Where $u=(1,2,a),v_1=(1, 1, −1),v_2=(2,−1,0),v_3=(3,0,−1)$
So we have this, and now we are given that $G◦F = 0$ which means
$$G(F(x)) = 0$$
Is my approach correct? If so how do I proceed?