# Winter 2017 MATH 504: Abstract linear algebraBranko Ćurgus

Monday, March 6, 2017

Monday, February 20, 2017

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

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

Wednesday, January 4, 2017