Challenge Problems for SSEA 51, Homework 3
(1) On page 42 of Levandosky there is a list of all possible 2-by-3 reduced row echelon
form matrices. Using the same terminology (that is, an * denoting any real number),
write down a list of all possible 3-by-2 reduced row echelon form matrices, and also
write down a list of all possible 3-by-3 reduced row echelon form matrices.
(2) Let A and B be two m × n matrices, and suppose that rref(A) = rref(B). Prove
that dim(C(A)) = dim(C(B)) [even though C(A) need not equal C(B)] and that
dim(N (A)) = dim(N (B)).
(Hint: Proposition 11.2 may be helpful.)
(3) Consider a generic 2-by-2 matrix:
w x
A=
.
y z
There are four possibilities for rref(A), where ∗ denotes any real number:
0 0
0 1
1 ∗
1 0
,
,
, and
.
0 0
0 0
0 0
0 1
For each possible form, give algebraic conditions on w, x, y, and z that are necessary
and sufficient to ensure that rref(A) is of that form. For example, the first form arises
if and only if w = 0, x = 0, y = 0, and z = 0.
(4) Tie three pieces of string into three loops (or draw a picture of the loops) so that:
(a) The three loops are linked. That is, you can’t pull one loop arbitrarily far away
from the others.
(b) After cutting any one of the three loops, the remaining two loops are not linked.
That is, you can pull the two remaining loops arbitrarily far apart.
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