Math 108a
Professor: Padraic Bartlett
Lecture 10: Range and Null Space, part 2
Week 4 UCSB 2013
In our last few classes, we've discussed the two concepts of range and null space. In particular, the last set of notes that we put online ended with a discuss
Math 108a
Professor: Padraic Bartlett
Lecture 9: Range and Null Space
Week 3 UCSB 2013
Today's lecture is on the concepts of range and null space, a pair of concepts related to the linear maps we've been studying recently. We define these two objects belo
Math 108a
Professor: Padraic Bartlett
Lecture 8: Linear Maps
Week 3 UCSB 2013
In our first week, we studied fields and vector spaces as abstract objects. In our second week, we transitioned from this abstract view to a more concrete and hands-on approach:
Math 108a
Professor: Padraic Bartlett
Lecture 7: Dot Products
Week 3
UCSB 2013
This talk is designed to introduce the dot product. We do this below:
1
Dot Products
Given a pair of vectors in Rn , the dot product operation is the following map:
Denition. T
Math 108a
Professor: Padraic Bartlett
Lecture 6: Basis and Dimension
Week 2
UCSB 2013
In our last talk, we introduced the concepts of span and linear independence. We
continue introducing new vector space concepts with todays pair of denitions: the concep
Math 108a
Professor: Padraic Bartlett
Lecture 5: Span and Linear Independence
Week 2
UCSB 2013
Our lectures thus far have focused on the two concepts of elds and vector spaces.
In specic, weve studied these two objects in a fairly formal setting: weve cre
Math 108a
Professor: Padraic Bartlett
Lecture 4: Subspaces
Week 1
UCSB 2013
In our last class, we introduced the formal denition of a vector space: i.e. a set V
along with a eld F such that the following properties hold:
Closure(+): v, w V, we have v +
w
Math 108a
Professor: Padraic Bartlett
Lecture 3: Vector Spaces
Week 1
UCSB 2013
The main reason we studied elds in our rst lecture is not because were particularly
interested in the denitions and concepts of elds themselves. (Which is not to say that
elds
Math 108a
Professor: Padraic Bartlett
Lecture 2: Fields, Formally
Week 1
UCSB 2013
In our rst lecture, we studied R, the real numbers. In particular, we examined how the
real numbers interacted with the operations of addition and multiplication, listing a
Math 108a
Professor: Padraic Bartlett
Lecture 1: Fields, Informally
Week 0
1
UCSB 2013
The Real Number System
The real numbers, denoted R, have a lot of dierent denitions. The most common is
probably the innite decimal sequence denition, which we state he
Math 108a
Professor: Padraic Bartlett
Practice Homework 5: Injection, Surjection, and Linear Maps
Not due: just for practice
UCSB 2013
Have fun!
1. Let T : U V and S : V W be a pair of injective maps. Dene the composition
of these two maps S T : U W as th
Math 108a
Professor: Padraic Bartlett
Homework 4: Properties of Vector Spaces
Due Thursday, Oct. 24, 3pm, South Hall 6516
UCSB 2013
Remember: homework problems need to show work in order to receive full credit. Simply
stating an answer is only half of the
Math 108a
Professor: Padraic Bartlett
Homework 3: Properties of Vector Spaces
Due Thursday, Oct. 10, 3pm, South Hall 6516
UCSB 2013
Remember: homework problems need to show work in order to receive full credit. Simply
stating an answer is only half of the
Math 108a
Professor: Padraic Bartlett
Homework 2: Vector Spaces
Due Thursday, Oct. 10, 3pm, South Hall 6516
UCSB 2013
Remember: homework problems need to show work in order to receive full credit. Simply
stating an answer is only half of the problem in ma
Math 108a
Professor: Padraic Bartlett
Lecture 11: Understanding Null Space
Week 4
UCSB 2013
In our last lecture, we started studying the motivation behind the concept of the null
space. In todays talk, we return to this study.
1
Null Space: The Theorem
In
Math 108a
Professor: Padraic Bartlett
Lecture 12: Injection, Surjection and Linear Maps
Week 4
UCSB 2013
Todays lecture is centered around the ideas of injection and surjection as they relate
to linear maps. While some of you may have seen these terms bef
Math 108B Intro to Linear Algebra Winter 2010
Professor: Kenneth C. Millett
Office: 6512 South Hall
Office Hours: R 8:30 11:00
Email: [email protected]
Graduate Assistant: Tomas Kabbabe
Office: 6432K South Hall
Office Hour: W 10:00 11:00
Email: [email protected]
Solutions to Axler, Linear Algebra Done Right 2nd Ed.
Edvard Fagerholm [email protected]_helsinki.|gmail.com
Beware of errors. I read the book and solved the exercises during spring break (one week), so the problems were solved in a hurry. However, if
MATH 108B HW 2 SOLUTIONS
RAHUL SHAH
Problem 1. [6.27]
Solution. Given z = (z1 , . . . zn ), z = (z1 , . . . zn ) Fn , notice that
n
z, z
n
=
zi ei ,
i=1
zi e i
i=1
n
n
=
zi ei ,
i=1
zj ej
j =1
n
zi zj ei , ej
=
i,j =1
n
zi zi
=
i=1
Now dene T (z1 , . . .
MATH 108B HW 3 SOLUTIONS
RAHUL SHAH
Problem 1. [7.11]
Solution. Assume T is a normal operator on a complex vector space. Then, by the complex spectral theorem, we nd
that T has an orthonormal basis consisting of eigenvectors. Thus M(T ) is a diagonal matr