Answer
a. True
b. True
c. False
d. False
e. False
Work Step by Step
a. Every 2 vectors who are not scalar multiples of each other form a linearly independent set in $R^2$, but their dot products are not necessarily 0. For instance, take $\begin{bmatrix}
1\\
2\\
\end{bmatrix}$ and $\begin{bmatrix}
1\\
3\\
\end{bmatrix}$. This is true for all $R^n$.
b. Weights can be calculated by$ \frac{y\cdot u_i}{u_i\cdot u_i}$ for each vector in the orthogonal basis.
c. Normalizing a vector is the same as taking a scalar multiple of it. This does not change the fact that th vectors are orthogonal because $c\vec{u}\cdot d\vec{v}$ is still 0 as long as $\vec{u}\cdot \vec{v}=0$ regardless of c and d.
d. Only square matrices are orthogonal.
e. $||y-\widehat{y}||$ gives the distance from y to L.
