Here is the problem from Ex 8.2 in Hoffman and Kunze
Part $a$, $b$ and $c$ were trivial to solve but I am not able to solve $d$ part. The formula for part $a$ came out to be $$E(x_1,x_2) = \frac{(3x_1 + 4x_2)}{25} (3,4)$$ Now, when I try to solve $d$ part, I thought of simply assuming one basis vector to be $(x_1,x_2)$ and equating to $(1,0)$ but it turns out to be unsolvable. What am I do wrong? Can someone please help? I am reading the book on my own.
Best Answer
For part d let's think about what exactly that means. We want an orthonormal basis $\{v,w\}$ such that $E$ is $\begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}$. This is the same thing as saying that $E(v)=v$ and $E(w) = 0$. Well by definition of a projection, anything in the image of $E$ satisfies $E(v)=v$. We want an orthonormal basis, so let's take this to be unit length. For instance, we can take $v=(3/5,4/5)$.
Now we need some nonzero $w$ with $E(w)=0$ and $v\perp w$. That will automatically imply that $v$ and $w$ are linearly independent, giving us our desired basis. Now, $E$ is an orthogonal projection, so its kernel is the orthogonal complement of its image. Thus, we need only find some unit length vector perpindicular to $(3/5,4/5)$. We can take $w=(-4/5,3/5)$.