Advanced Linear Algebra (Math 424/524) Review Exercises

September 18, 2002

1. Find the reduced row echelon form of the matrix

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

2. Find the determinant of the 3 × 3 matrix

( \begin{matrix}0 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 0 \end{matrix} ) .

3. Find the inverse of the orthogonal matrix

\frac{1}{7} ( \begin{matrix}2 & 3 & 6 \\ 6 & 2 & −3 \\ 3 & −6 & 2 \end{matrix} ) .

4. Let T be the linear transformation from R^{3} to R^{2} given by

T (x_{1}, x_{2}, x_{3}) = (3x_{2} − x_{3}, x_{1} + 4x_{2} + x_{3}) .

Find the unique 2 × 3 matrix A such that

T (x) = Ax

for each x in R^{3}.

5. Find a basis for the vector subspace of R^{4} that consists of all solutions of the system of linear equations

x_{1} − 2x_{3} + x_{4} = 0
x_{2} + 3x_{3} − 2x_{4} = 0

6. Let f be the linear function from R^{4} to R^{4} that is defined by f(x) = Mx where M is the matrix

( \begin{matrix}1 & −1 & −2 & 0 \\ − 1 & 2 & 0 & −3 \\ 2 & 0 & −1 & 1 \\ 0 & −1 & 2 & 3 \end{matrix} ) .

(a) Find a basis of the kernel of f.

(b) Find one or more non-redundant linear equations that characterize the image of f, i.e., equations for which

the set of common solutions is the image of f.

7. Give an explicit description of the set of all n × n matrices that are similar to the n × n identity matrix.

8. Let g be the linear function from R^{3} to R^{3} that is defined by g(x) = Rx where R is the matrix

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

Find as many as possible non-parallel eigenvectors of g, i.e., non-zero vectors x in R^{3} for which g(x) is a scalar multiple of x.