Here is a review of some (but not all) of the topics you should know for the
midterm. These are things I think are important to know. I haven’t seen the
test, so there are probably some things on it that I don’t cover here. Hopefully
this covers most of them.
•
Vector Spaces
Review properties on Shankar page 2
Closure under multiplication: If

u
i
and

v
i ∈
V
,
then
a

u
i
+
b

v
i ∈
V
for any
a, b
. Note when
b
= 0
this takes care of scalar multiplication also.
Inverses, identity, etc.
•
Linear independence
A set of vectors
{
v
i
i}
is linearly independent if
a

v
1
i
+
b

v
2
i
+
· · ·
= 0 has only one solution:
a
=
b
=
· · ·
= 0.
•
GramSchmidt procedure
If you have a set of linearly independent vectors

I
i
,

II
i
, . . .
you can always construct an orthonor
mal set of vectors as follows:

1
i
=

I
i
p
h
I

I
i

2
i
=

II
i  
1
ih
1

II
i
normalization constant

3
i
=

III
i  
1
ih
1

III
i  
2
ih
2

III
i
normalization constant
. . .
The normalization constants are chosen so that
h
2

2
i
=
1
,
h
3

3
i
= 1
, . . .
.
•
Basis
A basis of a vector space
V
is a set of vectors
{
v
i
i}
.
1
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View Full DocumentAny vector

u
i ∈
V
can be written in terms of these
vectors:

u
i
=
a

v
1
i
=
b

v
2
i
+
. . .
always has
a, b, .
. .
so that the equation is satisiﬁed.
•
Orthonormal (ON) basis
An ON basis is one for which
h
v
i

v
j
i
=
δ
ij
.
•
Decomposition of unity
If
{
v
i
i}
is an ON basis, then
∑
i

v
i
ih
v
i

=
I
.
•
Linear Operators
Linear operators have Ω
e
(
a

u
i
+
b

v
i
) =
a
Ω
e

u
i
+
b
Ω
e

v
i
.
•
Operator Inverses
The inverse of the product of operators is given by
the inverses of those operators in reverse order: (Ω
e
Λ
e
)

1
=
Λ
e

1
Ω
e

1
.
•
Commutators
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 Winter '03
 Nelson
 Linear Algebra, mechanics, Eigenvectors, Eigenvalues, basis, characteristic equation, Matrix representation

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