Exercise 6.2.19

Find an orthonormal basis for $W$ and use the formula in Theorem 6.6.

1.

Pick $\{\frac{1}{\sqrt{17}}(1,4)\}$ as a basis for $W$. Thus the orthogonal projection of $u$ is

$$⟨u,\frac{1}{\sqrt{17}}(1,4)⟩\frac{1}{\sqrt{17}}(1,4)=\frac{26}{17}(1,4).$$

2.

Pick $\{\frac{1}{\sqrt{10}}(-3,1,0),\frac{1}{\sqrt{5}}(2,1,0)\}$ as a basis for $W$. Thus the orthogonal projection of $u$ is

$$⟨u,\frac{1}{\sqrt{10}}(-3,1,0)⟩\frac{1}{\sqrt{10}}(-3,1,0)+⟨,\frac{1}{\sqrt{35}}(1,3,5)⟩\frac{1}{\sqrt{35}}(1,3,5)$$

$$=-\frac{1}{2}(-3,1,0)+\frac{4}{7}(1,3,5)=\frac{1}{14}(29,17,40).$$

3.

Pick $\{1,\frac{1}{\sqrt{3}}(2x-1)\}$ as a basis for $W$. Thus the orthogonal projection of $h$ is

$$⟨h,1⟩1+⟨h,\frac{1}{\sqrt{3}}(2x-1)⟩\frac{1}{\sqrt{3}}(2x-1)$$

$$=\frac{29}{6}+\frac{1}{18}(2x-1)=\frac{1}{9}(x+1).$$