Exercise 3

Consider the basis of $\real^3$ given by the vectors $$\vvec_1=\threevec2{-2}2,\hspace{24pt} \vvec_2=\threevec{-1}{-3}1,\hspace{24pt} \vvec_3=\threevec{2}0{-5}.$$

1. Apply the Gram-Schmit orthogonalization algorithm to find an orthonormal basis $\uvec_1$, $\uvec_2$, $\uvec_3$ for $\real^3$.

2. If $A$ is the $3\times3$ whose columns are $\vvec_1$, $\vvec_2$, and $\vvec_3$, find the $QR$ factorization of $A$.

3. Suppose that we want to solve the equation $A\xvec=\bvec = \threevec{-9}17$, which we can rewrite as $QR\xvec = \bvec$.

   1. If we set $\yvec=R\xvec$, the equation $QR\xvec=\bvec$ becomes $Q\yvec=\bvec$. Explain how to solve the equation $Q\yvec=\bvec$ in a computationally efficient manner.

   2. Explain how to solve the equation $R\xvec=\yvec$ in a computationally efficient manner.

   3. Find the solution $\xvec$ by first solving $Q\yvec = \bvec$ and then $R\xvec = \yvec$.