Advanced Linear Algebra Math 424/524

Assignment No. 3

Due November 13, 2002

1. Find a 2 × 2 real diagonal matrix D for which there exists a matrix U that is orthogonal relative to the standard inner product (the “dot” product) on R^{2} and satisfying

U^{t} ( \begin{matrix}1 & 2 \\ 2 & −1 \end{matrix} ) U = D .

2. Find an invertible 2 × 2 matrix U of rational numbers and a rational diagonal matrix D

such that

U^{t} ( \begin{matrix}1 & 2 \\ 2 & −1 \end{matrix} ) U = D .

3. When 2 ≠ 0 in the field F, find an invertible 2 × 2 matrix U in F such that

U^{t} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) U = ( \begin{matrix}1/9 & 0 \\ 0 & −1/9 \end{matrix} ) .

4. Find a 3 × 3 rational matrix that is orthogonal for the standard inner product on R^{3} with the property that none of its entries has absolute value 1.

5. Find an invertible matrix U such that

U^{t} ( \begin{matrix}0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) U = ( \begin{matrix}0 & 1 & 0 & 0 \\ − 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ) .

6. What conditions in the definition of inner product are not satisfied by the bilinear form Θ on the space M_{n}(R) of n × n real matrices defined by

Θ(M, N) = trace (MN) .

7. For a field F in which 2 = 0 give an example of a bilinear form on F^{3} that is skew-symmetric but not alternating.

8. Let b be the bilinear form on F^{3} given by

b(x, y) = x^{t} ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} ) y .

(a) Explain very briefly why b is dualizing.

(b) Find the left orthogonal complement of U, i.e.,

{v ∈ F^{3} | b(v, u) = 0 for each u ∈ U}

in each of the three cases when U is a coordinate axis.

(c) When 2 ≠ 0 in F, find a symmetric bilinear form s and an alternating bilinear form a on F^{3} such that b = s + a.