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 ⊂ R
4 be the space of solutions of the system of linear
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
(1, 2, 0)t
,(2, 1, 2)t
,(3, 1, 1)t
is a basis of R
(b) Find the coordinate vector of the vector v = (1, 2, 3)t with respect to
(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 = F
(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
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