189-251B: Algebra 2 Assignment 3

Due: Wednesday, January 29.

1. Solve the linear equation Ax = y over the field Z_{2}, where

A = ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{matrix} ) , y = ( \begin{matrix}0 \\ 1 \\ 1 \end{matrix} ) .

2. Solve the linear equation Ax = y over the field Z_{2},

A = ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} ) , y = ( \begin{matrix}0 \\ 1 \\ 1 \end{matrix} ) .

How many solutions does this equation have? Write them all down.

3. Show that the number of distinct solutions of a system of linear equations (in any number of equations, and unknowns) over the field Z_{p} is either 0, or a power of p.

4. Show that there is no error-correcting code of dimension 5 in Z_{2}^{7}, so that the example constructed in class is in some sense optimal, in the sense that it is the error-correcting code of largest possible dimension in Z_{2}^{7}.

5. What is the largest possible dimension of a linear code in Z_{2}^{15} that can detect and correct a single (i.e., one-bit) error?

6. Let V be the vector space of polynomials of degree at most 5 over the field R of real numbers, and let T be the linear transformation from V to V defined by

T (f) = \frac{d^{3}}{dx^{3}} f + \frac{d^{2}}{dx^{2}} f.

Describe the kernel of T, and its image. What are the dimensions of these subspaces? What is the subspace of V generated by ker(T) and Image(T)?

7. A linear transformation T : V ⟶ V is called a projection, or an idempotent, if it satisfies T^{2} = T. Show that V can be expressed as the direct sum of ker(T) and Image(T).

8. Let V be a vector space over a field F. A linear transformation N : V ⟶ V is said to be nilpotent if N^{k} = 0 for some k. Show that if N is nilpotent, then the linear transformation 1−N (where 1 denotes the identity transformation) is invertible. (Hint: think of the power series expansion for \frac{1}{1−x} when x < 1.)