Homework 6, due 10/2/2015
1. (Artin 3.4.1) Find a basis for the space of n × n symmetric matrices (those
for which At = A).
2. (Artin 3.4.2) Let W ⊂ R4 be the space of solutions of the system of lienar
equations AX = 0, where
2 1 2 3
.
1 1 3 0
Find a basis for W .
3. (Artin 3.4.4) Let A be an m × n matrix, and let A0 be the result of a
sequence of elementary row operations on A. Prove that the rows of A
span the same space as the rows of A0 .
4. (Artin 3.5.1)
(a) Prove that the (ordered) set
B = (1, 2, 0)t , (2, 1, 2)t , (3, 1, 1)t
is a basis of R3 .
(b) Find the coordinate vector of the vector v = (1, 2, 3)t with respect to
this basis.
(c) Let
B0 = (0, 1, 0)t , (1, 0, 1)t , (2, 1, 0)t .
Determine the change-of-basis matrix P from B to B0 .
5. (Artin 3.5.4) Let Fp be the field of p elements for some prime p, and let
V = F2p . Prove:
(a) The number of bases of V is equal to the order of GL2 (Fp ).
(b) The order (i.e. size) of GL2 (Fp ) is p(p + 1)(p − 1)2 , and the order of
SL2 (Fp ) is p(p + 1)(p − 1). (Recall that SL2 (F ) is the subgroup of
GL2 (F ) of matrices over a field F having determinant 1.)
6. (Artin 3.5.5) How many subspaces of each dimension are there in
(a) F3p ?
(b) F4p ?
1