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Math 260, Spring 2012
Jerry L. Kazdan
Class Outline: Jan. 19, 2012
1.
Definitions:
Homogeneous equation, inhomogeneous equation.
Basic Lemma:
If you have
n
linear algebraic equations in
k
unknowns, and if
n > k
(so more equations than unknowns), then the homogeneous equations always have at
least one nontrivial solution (
nontrivial
means a solution other than 0). In other
words, the dimension of the
nullspace
of the corresponding map is greater than 0.
Definitions:
For a linear map
L
:
V
→
W
(here
V
and
W
are linear spaces):
The
image
of
L
is the set of all
y
∈
W
for which
Lx
=
y
has a solution.
The
nullspace
or
kernel
of
L
is the set of all solutions of the homogeneous equation
Lx
= 0.
Notation:
I
(
L
),
N
(
L
), ker(L).
2. Maps:
f
:
S → T
.
Definitions:
onetoone (injective),
onto (surjective),
invertible (bijective)
3.
A
:
R
2
→
R
2
If
A
is linear, it maps straight lines to straight lines, and preserves
parallelism
Step 0: What is a straight line?
Step 1: What about degeneracies like
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 Spring '12
 STAFF
 Linear Algebra, Algebra, Equations

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