Winter 2017
MATH 504: Abstract linear algebra
Branko Ćurgus

Monday, March 6, 2017

I updated my notes on Inner Product Spaces.

Monday, February 20, 2017

These are my notes on Inner Product Spaces.

Saturday, February 4, 2017

Before proceeding with our next topic we reviewed the Fundamental Theorem of Algebra. Here are my notes with a simple proof of the Fundamental Theorem of Algebra.
Our next topic is Eigenvalues and Eigenvectors of linear operators on finite dimensional complex vector spaces.

Tuesday, January 24, 2017

Here are my notes on Linear operators between vector spaces.

Monday, January 23, 2017

Here is a list of recommended exercises from the textbook: Chapter 1: 3-7, 10, 12-15. Chapter 2: 9, 11-16.

Tuesday, January 10, 2017

I updated my notes on Vector spaces.
I post my notes entitled Bases which deal with topics covered in Chapter~2 of the book.

The Assignment 1 consists of the following five problems.

Problem 1. Problem 7.2 in my notes on Vector spaces.
Problem 2. Problem 7.3 in my notes on Vector spaces. The easiest way to do this problem is to represent each $\mathcal S_\gamma$ as a span. Your proof could start as follows: I claim that $\mathcal S_\gamma = \operatorname{span}\bigl\{ \ldots \bigr\}$. Here is a proof of my claim. (Write your proof.)
Problem 3. Problem 7.4 in my notes on Vector spaces.
Problem 4. Problem 7.7 in my notes on Vector spaces. Before doing this problem solve Problem 7.5 and internalize my solution of Problem 7.6.

Problem 5. Let $\mathcal V$ be a vector space over $\mathbb F$. Let $\mathcal A$ be a linearly independent subset of $\mathcal V$. Let $u \in \mathcal V$ be arbitrary. By $u + \mathcal A$ we denote the set of vectors $\{u+v : v \in \mathcal A \}$.

(a)	Prove the following implication. If $w \notin {\rm span}\, \mathcal A$, then $ w + \mathcal A$ is a linearly independent set.
(b)	Is the converse of the implication in (a) true? Justify your claim.
(c)	Let $\alpha_1,\cdots,\alpha_n \in \mathbb F$, let $v_1,\ldots, v_n$ be distinct vectors in $\mathcal A$ and let $w = \alpha_1 v_1 + \cdots + \alpha_n v_n$. Find a necessary and sufficient condition (in terms of $\alpha_1, \ldots, \alpha_n$) for the linear independence of the vectors $v_1 + w, \ldots, v_n + w$.

Thursday, January 5, 2017

My notes of Chapter 1; the title of the notes is Vector spaces.

Wednesday, January 4, 2017

Winter 2017 MATH 504: Abstract linear algebra Branko Ćurgus

Winter 2017
MATH 504: Abstract linear algebra
Branko Ćurgus