Linear Algebra (Math 220)
Assignment due Tuesday, April 15

1.  Preparation

Relevant Reading:

Course notes on “change of basis” (also available as PDF)

Lay § 4.7

Hefferon § 3.V

2.  Exercises

1. Find the determinant of the $4\times 4$ matrix $$\left(\begin{array}{cccc}\hfill 1& \hfill 2& \hfill -4& \hfill 1\\ \hfill -2& \hfill 10& \hfill -1& \hfill 1\\ \hfill 1& \hfill 0& \hfill 1& \hfill 5\\ \hfill 2& \hfill -9& \hfill 1& \hfill 0\end{array}\right)\text{.}$$

2. Express the $3\times 3$ matrix $$\left(\begin{array}{ccc}\hfill 2& \hfill 2& \hfill 1\\ \hfill -2& \hfill 1& \hfill 2\\ \hfill -1& \hfill 2& \hfill -2\end{array}\right)$$ as a product of elementary matrices.

3. Find the matrix with respect to the standard basis of ${\mathbf{R}}^2$ of the reflection in the line through the origin that has angle of elevation $\theta ⁄2$ (counterclockwise from the positive direction along the first coordinate axis).

4. Find the matrices for change of basis in both directions between the standard basis of ${\mathbf{R}}^3$ and the basis formed by the columns of the matrix $$\left(\begin{array}{ccc}\hfill 3& \hfill 2& \hfill 4\\ \hfill 2& \hfill -3& \hfill 1\\ \hfill 3& \hfill -6& \hfill 1\end{array}\right)\text{.}$$

5. Let $f$ be the linear function from ${\mathbf{R}}^3$ to ${\mathbf{R}}^3$ given by $f\left(x\right)=Mx$ where $$M=\left(\begin{array}{ccc}\hfill 1& \hfill 5& \hfill -2\\ \hfill -2& \hfill 4& \hfill -3\\ \hfill -1& \hfill -3& \hfill 1\end{array}\right)\text{.}$$ Find the matrix of $f$ relative to the basis of ${\mathbf{R}}^3$ given by the columns of the matrix in the preceding exercise.