MATH 25A, 9/19/14 LECTURE NOTES
BASES
Recap. For any subset S V ,
n
span(S) =
ai si n N0 , ai F, si S
W =
W V subspace
SW
i=1
Denition 0. S V is called generating if V = span(S).
Proposition 1. Let V be an F -vs and S V be a subset. TFAE:
1) For any s S,
Math 25a
Steven Ban
Notes
October 31, 2014
Dention: [Halloween Denition]
Group G operateson set X, means homomorphism G Aut(X), where Aut(X) are
bijective endomorphisms.
Remark: Group operations are the same as Group actions
Let V be n-dimensional F-vecto
Math 25a
Steven Ban
Notes
November 3, 2014
Theorem 5 [Continued from last class]Start with step 3 of last classs proof:
Let V be an n dimensional F vector space and f L(V, V ). Assume that f = (c1
X)n1 . . . (cn X)nk
Recall the generalized eigenspace (Ec
Math 25a
Steven Ban
Notes
November 5, 2014
RIP Jordan Normal Form
Let V be a nite dimensional F vector space. Recall:
g L(V, V ) is nilpotent if g k = 0 for some k N . (Def 1)
Let g L(V, V ), v0 V, k N such that vk = g k (v0 ) = 0 but vk+1 = g k+1 (v0 )
Math 25a
Problem Set 10
Steven Ban
November 7, 2014
PARENT WEEKEND!:
Tasho will be out of town on Monday. Bad news: class is still happening and oce hours
are not happening. Good news: we get a one day extension on the PSET (as if that wasnt
happening alr
Math 25a
Problem Set 10
Steven Ban
November 10, 2014
feat. Guest/Substitute Lecturer
First order of busines, extend the denition of inner product spaces to C = cfw_a + bi|a, b
R. We dene the complex conjugate as a + bi = abi with the properties that z +
Math 25a
Steven Ban
Notes
November 14, 2014
Let (V, < / >) be an inner product space.
Theorem 1.
1. Cauchy-Schwartz: For all v, w V , we have
| < v, w > | |v| |w|
Equality occurs when v and w are linearly dependent
2. Triangle Inequality: For all v, w V ,
Math 25a
Problem Set 10
Steven Ban
November 12, 2014
Survey Notes
Were sticking with the new grading system (splitting up the problems and giving them
to dierent graders)
Tasho will stay at the teaching pace hes going at right now
Tasho will also relea
Math 25a
Steven Ban
Notes
November 19, 2014
Let (V, < / >) be an inner product space over F is either R or C.
Recall:
For any f L(V, V ), there exists a unique f L(V, V ) such that < f (v), w >=<
v, f (w) > for all v, w V .
If f = f then f is called sel
Math 25a
Steven Ban
Notes
October 29, 2014
Note: Were no longer cutting Jordan Normal forms and the spectral theorem. Were now
cutting symmetric bilinear forms, hermetian forms, and etc. On that note, lets start Jordan
Normal Form
Let V be an n dimensiona
Math 25a
Steven Ban
Notes
October 27, 2014
Let V be a nite dimenional F-vector space f L(V, V )
Recall
1. For c F , Ec (f ) = cfw_v V |f (v) = cv is our f-invariant subspace
2. c is an eigenvalue Ec (f ) = cfw_0
Fact 1. Ec (f ) = N (f c idV )
Corallary 2.
MATH 25A, 9/22/14 LECTURE NOTES
FINITE-DIMENSIONAL SUBSPACES
Recall. B V is called a basis if it is both LI and generating. If B1 , B2 V
are bases, and #B1 < 1, then #B2 < 1 and #B1 = #B2 .
Denition 1. An F -vector space V is called nite-dimensional if th
MATH 25A, 9/26/14 LECTURE NOTES
ISOMORPHISMS
Recall. A map f : V ! W between F -vs is linear if
f (v + w) = f (v) + f (w)
f (c v) = c f (v)
for v, w 2 V, c 2 F .
If (v1 , . . . , vn ) is a basis of V , then
cfw_set of all linear maps f : V ! W ! W n , f
MATH 25A, 9/24/14 LECTURE NOTES
LINEAR MAPS
groups
f :GG
elds
is called a (homo)morphism
Denition 1. A map f : F F between
f :V V
F -vector spaces
if
f (g h) = f (g) f (h)
for all g, h G
f (a b) = f (a) f (b), f (a + b) = f (a) + f (b), f (1) = 1 for all
MATH 25A, 9/29/14 LECTURE NOTES
COORDINATE ISOMORPHISMS
Recall.
F n = F . . . F = cfw_(a1 , . . . , an ) | ai F . From now on, write elements
of F n vertically, i.e.
a1
.
F n = . ai F = M at(n, 1; F )
.
an
M at(n, m; F ) L(F m , F n )
=
canon
c1
c
Math 25a
Steven Ban
Notes
October 20
DETERMINANTS
Let V be a nite dimensional F-vector space.
Recall
Altn (V ; F ) is the space of alternating n-linear forms on V .
dimAltn (V ; F ) = 1
Fact 1. Let X be a 1 dimensional F-vector space. Then
1. For any x,
MATH 25A, 10/1/14 LECTURE NOTES
DUAL SPACES
Example. V = F n . For 1 i n let e : F n F be the unique linear map
i
dened by
e (ej ) = ij
i
In other words,
a1
.
e i . = ai
.
an
Then for v, w V ,
v = 0 e (v) = 0 for all i
i
v = w e (v) = e (w) for all i
Math 25a
Steven Ban
Notes
October 22,2014
Recall
If X is a 1 dimensional F vector space, then
L(X, X) F
=
a idv a
or L(X, X) is ismorphic to F (since there always exists the identity map from X to
X, we have a basis)
If V is a n dimensional F vector spa
Math 25a
Steven Ban
Notes
October 24, 2014
EIGENVALUES
Let V be an n-dimensional F vector space. Now consider endomorphisms on V . The
simplest endomorphism is idv : V V . Equally simple is c idV : V V . Then [c idv ] is
simply a diagonal matrix with c al
Math 25a
Steven Ban
Notes
November 17, 2014
Spectral Theorem Revisiting the question: When is an endomorphism diagonalizable?
cfw_Tasho: For the Spectral theorem, the material is so basic that Axler and HomanKunz cover the material well, and Lax covering