M260. Homework 1
Prove the following propositions.
Proposition 1. Let f : A B be a function between two sets A and B.
Show that followings are true
1. f is injective if and only if b B, the set (f 1 (b) contains at most one
element.
2. f is surjective if
Math 260
Exam 3
May 10, 2013
Answer the questions in the spaces provided on the question sheets. Please write in clear
handwriting. Show all work.
Name:
1. Write the definition of following terms:
(a) (3 points) An inner product space V over a field K. An
Set Theory review for Math 260
Kwang Cha
I will introduce more set notations and definitions as we proceed through
the course. This note gives the absolute minimum you have to know for this
course. Also, I have assumed that you already know the definition
Math 260
Final Exam
July 13, 2016
Sample Solution for Questions 3 and 4
1. Write the following definitions.
(a) (1 point) A vector space V over a field K.
(b) (1 point) A linear map f : V W between vector spaces V and W .
(c) (1 point) The image and the k
A Short Note on finitely generated vector spaces
May 6, 2016
1
Notations.
Throughout this note, V will denote a vector space over a field K.
2
Generating set of a vector space.
Proposition 1: Let be the set of all subspaces of V . If is some nonmepty
subs
Math 260
Final Exam
May 20, 2013
Answer the questions in the spaces provided on the question sheets. Please write in clear
handwriting. Show all work.
Name:
1. (a) Write the definitions of following terms:
i. (2 points) A vector space V over a field K.
ii
Math 260
Exam 1
June 17, 2013
Answer the questions in the spaces provided on the question sheets. Please write in clear
handwriting. Show all work.
Name:
1. Write the definition of following terms:
(a) (3 points) A vector space V over a field K.
(b) (1 po
Quotient Space
Kwang Cha
Definition. Let M be a vector space over K. Let N be a subspace of M .
The quotient space of M by N , denoted by M/N , is a set cfw_x + N |x M .
Remark. Recall that x + N is a set cfw_x + n|n N . So M/N is a set of
subsets of M .
Math 260
Exam 2
April 16, 2013
Answer the questions in the spaces provided on the question sheets. Please write in clear
handwriting. Show all work.
Name:
1. (a) (5 points) Compute following operation.
3
2
4 3
1 0 1
1 6 + 2 4 =
.
1 1 0
2 1
1
2
(b) (5 p
Math 260
Exam 2 info
Summer, 2016
This sheet is general overview of what is going to asked in exam 2. The lecture note on
determinant can be found on Blackboard.
1. Definitions
(a) An inner product space.
(b) Norm of a vector in an inner product space.
(c