Advanced Linear Algebra (Math 424/524)

8 September 2002

These are just exercises to aid in reviewing your previous knowledge of linear algebra.

1. Find a parametric description of the set of all solutions (x, y, z) of the system of linear

equations

\begin{matrix}x − 2y + z = 0 \\ 2x − 3y − z = 6 \end{matrix} .

2. Find the inverse of the matrix

( \begin{matrix}1 & −2 \\ 1 & −3 \end{matrix} ) .

3. Let M be the 3 × 3 matrix that is given by

M = ( \begin{matrix}0 & 2 & 4 \\ 1 & 0 & 1 \\ 3 & 1 & 0 \end{matrix} ) .

Find the determinant of M.

4. Let

f : R^{3} ⟶ R^{3}

be the map defined by f(x) = Mx, where M is the matrix

M = ( \begin{matrix}0 & 2 & 4 \\ 1 & 0 & 1 \\ − 2 & 1 & 0 \end{matrix} ) .

(a) What is the rank of the matrix M?

(b) Find an equation for the image of f.

(c) Find a parametric representation of the fiber of f over the point (6, 2, −1).

(d) Find a point p in the image of f such that the vector drawn from the origin to p is perpendicular to the vector drawn from the origin to (6, 2, −1).