Linear Algebra with Applications, 10th Edition

Section 7.5 Exercises

1. For each of the following vectors x, find a rotation matrix R such that $R\mathbf{x}={\left|\right|\mathbf{x}\left|\right|}_2{\mathbf{e}}_1$:

   1. $\mathbf{x}={(1,1)}^T$

   2. $\mathbf{x}={(\sqrt{3},-1)}^T$

   3. $\mathbf{x}={(-4,3)}^T$

2. Given $\mathbf{x}\in {ℝ}^3$, define

   $$\begin{array}{cc}{r}_{\mathit{\text{ij}}}={(x_i^2+x_j^2)}^{1/2}& i,j=1,2,3\end{array}$$

   For each of the following, determine a Givens transformation $G_{ij}$ such that the ith and jth coordinates of $G_{ij}\mathbf{x}$ are $r_{ij}$ and 0, respectively:

   1. $\mathbf{x}={(3,1,4)}^T,i=1,j=3$

   2. $\mathbf{x}={(1,-1,2)}^T,i=1,j=2$

   3. $\mathbf{x}={(4,1,\sqrt{3})}^T,i=2,j=3$

   4. $\mathbf{x}={(4,1,\sqrt{3})}^T,i=3,j=2$

3. For each of the given vectors x, find a Householder transformation that zeroes out the last two entries of the vector.

   1. $\mathbf{x}={(-1,8,-4)}^T$

   2. $\mathbf{x}={(3,6,2)}^T$

   3. $\mathbf{x}={(0,-3,4)}^T$

4. For each of the following, find a Householder transformation that zeroes out the last two coordinates of the vector:

   1. $\mathbf{x}={(5,1,4,8)}^T$

   2. $\mathbf{x}={(4,-3,-2,-1,2)}^T$

5. Let

   $$A=\left[\begin{array}{ccc}1& \hfill 3& \hfill -2\\ 1& \hfill 1& \hfill 1\\ 1& \hfill -5& \hfill 1\\ 1& \hfill -1& \hfill 2\end{array}\right]$$

   1. Determine ...