2
votes
2answers
225 views

Fundamental theorem of linear algebra

When I studied linear algebra we (our books, our professors) used to call Fundamental theorem of linear algebra the theorem that says: Fundamental theorem of linear algebra: A linear ...
10
votes
8answers
296 views

$2\times2$ matrices are not big enough

Olga Tausky-Todd had once said that "If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this." There are, however, assertions about matrices that are true for ...
26
votes
6answers
1k views

Cool mathematics I can show to calculus students.

I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen. Every discussion section I am trying to show my students (give them as a ...
7
votes
4answers
210 views

A Handwaving Proof of a Specific Existence and Uniqueness Theorem

My problem is as follows: Given the second order homogeneous linear differential equation with constant coefficients $$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+c\,y(x)=0,$$ is there a good heuristic ...
0
votes
1answer
95 views

Graphical Demonstration of Linear Transformations on $\mathbb{R}^2$

I'm looking for some applet, program, software, demonstration etc. to use it in a class while teaching linear transformation in $\mathbb{R}^2$, so that the students can graphically understand the ...
9
votes
2answers
386 views

Etymology of the word “normal” (perpendicular)

While the word "normal" is one of the most overloaded mathematical terms, in linear algebra, it is usually associated with the notion of being perpendicular to something, as in "normal vector" or ...
8
votes
1answer
190 views

Pedagogy of Teaching the Inverse Matrix Method

I am teaching a group of (ordinary rather than honours) second-year engineers and we are studying matrices. I told the class today that as far as I could see we were only studying matrices and, ...
1
vote
0answers
62 views

What is the best way of introducing singular value decomposition (SVD) in a linear algebra course?

Why is it so important? Are there any applications which have a real impact?
8
votes
6answers
444 views

What are the technical reasons that we must define vectors as “arrows” and carefully distinguish them from a point?

I thought I'd bring this question to math.SE, as it could spark some interesting discussion. When I first learned vectors - a long time ago and in high school - the textbook and teachers would always ...
2
votes
5answers
4k views

Real life applications of general vector spaces

Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of ...
3
votes
1answer
192 views

What is a motivating way to introduce vectors?

What is a good way to introduce vectors on a linear algebra course so that students are motivated from the start? I need an opening which will have a real impact. Are there any motivating examples?
5
votes
5answers
661 views

Purpose of Linear Algebra

How much emphasizes should be on proof on a first course in Linear Algebra? I sometimes feel that they (proofs) crowd out a coherent vision for linear algebra. However I also think a central theme of ...
0
votes
1answer
466 views

Purpose of Inverse matrix

What use is the inverse matrix? I would not use it to solve linear systems but there must be some concrete or real life applications where it is used.
5
votes
3answers
249 views

Linear Algebra Journal

Is there any journal which has significant material on the teaching of linear algebra. I am investigating the most effective way to teach a course on Linear Algebra. What are the most important things ...
5
votes
0answers
194 views

Have Changes in Applications Made Linear Algebra More Central/Urgent?

In the days when my father taught civil engineering (some decades ago), mathematical applications seemed to be mainly "scientific." (This was the "space age.) Hence the most important branch of ...