LECTURE 4:
S U B S PA C E S A N D L I N E A R C O M B I N AT I O N S
Friday, October 2
1
last time
Last time we introduced the concept of a subspace of a vector space. Let U be
a subset of a vector space V. Then U is a subspace of V if U is a vector space
LECTURE 18:
E I G E N S PA C E S A N D T H E D I A G O N A L I Z AT I O N P R O B L E M
Monday, November 9
Throughout, let V denote an n-dimensional vector space over a field F.
1
computing the characteristic polynomial
To find the eigenvalues of a linear
LECTURE 11:
C O M P O S I T I O N A N D M AT R I X M U LT I P L I C AT I O N
Wednesday, October 21
1
last time
Last lecture we introduced ordered bases for a vector space and how one can use
such bases to write each vector uniquely in terms of the coordin
LECTURE 20:
T H E D I A G O N A L I Z AT I O N P R O B L E M I I I
Wednesday, November 18
Once again, let V denote an n-dimensional vector space over a field F.
Last time we introduced the eigenspace of a linear operator for a given eigenvalue and conside
LECTURE 5:
LINEAR SYSTEMS AND LINEAR INDEPENDENCE
Monday, October 5
1
last time
Last time we looked at a few examples of subsets of vector spaces and went
about proving that they were subspaces.
We then asked the question of what happens if a subset S of
LECTURE 9:
L I N E A R T R A N S F O R M AT I O N S I I
Wednesday, October 14
1
last time
Last time we introduced linear transformations T : V W of vector spaces.
Recall that these are simply functions from V to W, that preserves the vector
space structur
LECTURE 14:
I N V E RT I B I L I T Y A N D I S O M O R P H I S M S I I
Wednesday, October 28
1
isomorphisms
Definition 1.1. Two vector spaces V and W are said to isomorphic if there exists an
invertible linear transformation T : V W. We call the linear tr
LECTURE 16:
E I G E N V E C T O R S A N D E I G E N VA L U E S I
Wednesday, November 4
1
eigenvectors and eigenvalues
For the next several lectures we are going to focus our attention on linear
operators. Recall that if V is a vector space, then a linear
LECTURE 13:
I N V E RT I B I L I T Y A N D I S O M O R P H I S M S
Monday, October 26
1
invertibility of linear transformations
Definition 1.1. Let T : V W be a linear transformation of vector spaces. A linear
transformation U : W V is said to be the inve
LECTURE 10:
T H E M AT R I X R E P R E S E N TAT I O N O F A L I N E A R
T R A N S F O R M AT I O N
Monday, October 19
1
last time
Last time we introduced the range of a linear transformation. We showed that
this is a subspace of the vector space, and pro
LECTURE 6:
BASES AND DIMENSION I
Wednesday, October 7
1
last time
Last time we finished up Chapter 1 by introducing the notions of linear dependence / independence of a subset of vectors of a vector space.
2
overview
In this lecture were going to consider
LECTURE 8:
L I N E A R T R A N S F O R M AT I O N S I
Monday, October 12
1
last time
Last time we finished up our study of the basis of a vector space. Specifically,
we showed that every basis of a finitely vector space has the same number of
vectors. We
LECTURE 19:
E I G E N S PA C E S A N D T H E D I A G O N A L I Z AT I O N P R O B L E M
II
Friday, November 13
Throughout this lecture, let V denote an n-dimensional vector space over a field
F.
Last time we showed that a necessary condition for a linear
LECTURE 15:
DETERMINANTS
Monday, November 2
1
determinants
Definition 1.1. Let A be an n n matrix with entries in a field F. The i, j cofactor
i,j is pn 1q pn 1q matrix obtained by removing the ith row and jth
matrix A
column of A.
Example 1. Consider th
LECTURE 12:
M AT R I X M U LT I P L I C AT I O N A N D I N V E R T I B I L I T Y
Friday, October 23
1
last time
Last time we defined matrix multiplication by considering the matrix representation of the composition of two linear transformations.
2
overvie
LECTURE 21:
INNER PRODUCTS
Friday, November 20
Throughout these notes, F will denote either R or C.
So far in this course weve generalized the algebraic properties of Euclidean
vector spaces, but not the properties of the length of a vector or angle betwe
LECTURE 7:
BASES AND DIMENSION II
Friday, October 9
1
last time
Last time we considered subsets S of a vector space V that are both a spanning
set (or generating set) and also linearly independent. We call the sets a basis
for V.
Spanning
sets
Linearly
Ba
LECTURE 16:
E I G E N V E C T O R S A N D E I G E N VA L U E S I I
Friday, November 6
Last lecture we gave a method for determining the eigenvalues of a linear operator by considering its characteristic polynomial. Once we have the eigenvalues
of a linear
LECTURE 22:
N O R M S A N D O RT H O G O N A L I T Y
Monday, November 23
Throughout these notes, F will denote either R or C.
1
norms
Now that weve introduced the notion of an inner product, which generalizes
the notation of the dot product, we can use th
LECTURE 25:
ADJOINTS
Wednesday, December 2
Throughout this lecture, let V denote a finite dimensional inner product space
over the field F.
1
linear functionals
Definition 1.1. Let T P LpV, Fq. Then we call T a linear functional.
Example 1. Let V F3 and d
LECTURE 23:
THE GRAM-SCHMIDT PROCEDURE
Wednesday, November 25
Throughout these notes, F will denote either R or C.
Last time we introduced the definition of an orthonormal basis for an inner
product space and proved a result that illustrated why such a ba
LECTURE 24:
O RT H O G O N A L C O M P L I M E N T S
Monday, November 30
The following problem often arises in practice:
Given a subspace U of an inner product space V and a vector v P V,
find a vector u P U such that v u is as small as possible.
For exam
Math 115A-2: Homework 1
Due: January 8, 2015
1. Send me an e-mail introducing yourself. Let me know if you like to be called something
other than your registrar listing, and anything you think I should know about your background.
2. Read Sections 1.1-4 in
Math 115A Lecture 5, Fall 2015
Linear Algebra
Instructor: Gang Liu
Office: MS 6338
Phone:(310)206-1472
E-mail:gang at math.ucla.edu
Office Hours: MW 10:00-10:50 PM.
TA
Office Office Hours E-mail
Christopher Ohrt MS 3915
cohrt at ucla.edu
Textbook:
Friedbe
Math 115A Quiz 3
Length : 15 minutes
Instructions: Solve all 2 questions.
Books, notes, and electronic devices are not permitted.
Name:
What you would like to be called
1. Dene what is means for a function T to be a linear transformation.
2. Let T : V W b
Review Sheet - Midterm 2
Math 115A, Lecture 8, Fall 2015
The midterm will be held on Friday November 20 in MS 5147 from 3-3:50 pm.
The Exam will cover the material from sections 1.6, 2.1-2.5, and also from pages 245-247 in section
5.1 and page 261 in 5.2.
Math 115A Quiz 4
Length : 15 minutes
Instructions: Solve all 2 questions.
Books, notes, and electronic devices are not permitted.
When using previous results in your answers, make sure to accurately state what
they are.
Name:
What you would like to be cal
Math 115A Practice Midterm1 + Solutions
Instructions:
There are 5 problems. Make sure you are not missing any pages.
You may use without proof anything proven in the sections of the book or
in class. State clearly when you are using a theorem from class
Midterm2 - Math 115A, November 20, 2015
Name:
Signature:
What you would like to be called
Instructions:
There are 5 problems. Make sure you are not missing any pages.
You may use without proof anything proven in the sections of the book or
in class. Sta