ma511: Advanced Linear Algebra - Homework 5
QUESTION 1
Let F$_3$ = Z/3Z = {0, 1, 2} be the finite field with three elements. Let $V$ be a two-
dimensional F$_3$-vector space.
(a) How many elements are in the group GL$_2$(F$_3$)? Explain. using the formular $(p^n - 1)(p^n - p)$
(b) How many subspaces of $W \subseteq V$ are there? Explain.
(c) How many elements are in the set Hom$_F$(V, V)? Explain.
(d) How many elements of each rank are in Hom$_F$(V, V)? Explain.
(e) (Bonus!) Let $p$ be a prime number and $d$ a positive integer. Redo each question above
with the field replaced by F$_p$ = Z/pZ = {0, 1, 2, ..., p - 1} and the vector space $V$ a
d-dimensional F$_p$-vector space.
QUESTION 2
Let $V$ be a two-dimensional R-vector space with bases X = {$e_1, e_2$} and Y = {$f_1, f_2$}
related by the equations:
$f_1 = e_1 + e_2$
$f_2 = 3e_1 + 2e_2$.
Consider the linear operator T: V $\to$ V determined by:
T($e_1$) = $e_1 - e_2$
T($e_2$) = $e_1 + 5e_2$.
(a) Compute the two matrices $_X$[T]$_X$ and $_Y$[T]$_Y$.
