Course: Applied Linear Algebra

1. Consider vector space $V=\mathbb{R}^4$ with the standard dot product. Find the formula for the orthogonal projection $P_W(\vec v)$ when $\vec v=\begin{bmatrix}x \\y\\z\\w\end{bmatrix}\in\mathbb{R}^4$ and $W\subset \mathbb{R}^4$ is the line given by equations $$\begin{eqnarray} x+y+z+w &=&0 \\ x-y+z-w &=&0 \\ x+ 2y+3z+4w&=&0 \end{eqnarray} $$
2. Consider a vector space $V=C[-1,1]$ of continuous functions on the interval $[-1,1]$ with the inner product $\langle f,g\rangle=\int_{-1}^1 f(x)g(x)dx$. Find the orthogonal projection of function $f(x)=x$ onto the one-dimensional subspace $W=span\{\sin(\pi x) \}$. Note: The answer cannot be read out from the sub-Example at the bottom of page 5 of the notes! (But the same procedure can be applied.)