Exercise 2

Consider the vectors $$\wvec_1=\threevec111,\hspace{24pt} \wvec_2=\threevec{-1}01,\hspace{24pt} \wvec_3=\threevec1{-2}1.$$

1. Explain why these vectors form an orthogonal basis for $\real^3$.

2. Suppose that $A=\begin{bmatrix} \wvec_1 & \wvec_2 & \wvec_3 \end{bmatrix}$ and evaluate the product $A^TA$. Why is this product a diagonal matrix and what is the significance of the diagonal entries?

3. Express the vector $\bvec=\threevec{-3}{-6}3$ as a linear combination of $\wvec_1$, $\wvec_2$, and $\wvec_3$.

4. Multiply the vectors $\wvec_1$, $\wvec_2$, $\wvec_3$ by appropriate scalars to find an orthonormal basis $\uvec_1$, $\uvec_2$, $\uvec_3$ of $\real^3$.

5. If $Q=\begin{bmatrix} \uvec_1 & \uvec_2 & \uvec_3 \end{bmatrix}$, find the matrix product $QQ^T$ and explain the result.