Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [linear-algebra]

For questions related to linear mappings, matrices, basis, determinants, eigenvalues and other topics commonly done in a linear algebra course.

-4
votes
0answers
103 views

Should we relabel the slope formulas to prevent confusion with exponents? [on hold]

When you teach about slope in Algebra 1, should we relabel the slope formula to: $$m = \frac{y_b-y_a} {x_b-x_a}$$ and also temporarily relabel the slope-point formula to: $$y−y_a=m(x−x_a)$$ where ...
5
votes
0answers
74 views

How to create educational linear algebra animations?

I'm looking to create animations for a linear algebra course. I need things like writing and changing equations, including matrices, plotting of 2- and 3-dimensional axes with points, vectors, lines ...
5
votes
3answers
149 views

CoTeaching Elementary Linear Algebra

I and my colleague will teach an elementary linear algebra next few weeks, but the way our course is planned mostly is by turn teaching. By that I mean, my partner will teach the first 8 weeks of the ...
2
votes
1answer
116 views

Downloadable MCQs on Mathematics

I am looking for multiple choice question (MCQ) based tests on some Mathematics' topics (details below), which could be downloaded in most preferably tex (LaTex) format or doc/docx format. Kindly ...
10
votes
1answer
193 views

Analogies or explanations for duality, at the college sophomore level

This semester I taught the third semester of my college's freshman physics sequence. Nearly all the students are engineering majors. Compared to previous semesters when I've taught this course, I went ...
6
votes
1answer
99 views

Timing of when Cayley-Hamilton theorem is taught in Linear Algebra

I teach at a primarily undergraduate 4-year college in the US and we don't cover the proof of Cayley-Hamilton theorem in our linear algebra courses. I did however see both the computational and the ...
4
votes
4answers
320 views

Who actually uses $\mathbf i$, $\mathbf j$, $\mathbf k$ for the standard unit vectors?

I am wondering which research communities use the notation $\mathbf i$, $\mathbf j$, $\mathbf k$ for the three-dimensional unit vectors. The calculus textbook I have to use (Stewart) uses that ...
26
votes
9answers
6k views

Should college mathematics always be taught in such a way that real world applications are always included?

I am teaching Linear Algebra this semester with the textbook Introduction to Linear Algebra by Serge Lang and most (perhaps all?) my students are not majoring in mathematics. As I was carefully ...
10
votes
1answer
234 views

Is Lax's Linear Algebra and its Applications comprehensive or idiosyncratic?

I'm looking for a good abstract linear algebra text (i.e., not matrix crunching) for students who have completed a Strang-level linear algebra course plus exposure to a proof writing (e.g., induction ...
11
votes
1answer
320 views

Helping a student exasperated by abstract concepts in linear algebra

I am currently tutoring a student in linear algebra. She is a very hard worker and does well on computational problems, but struggles to build mathematical intuition. This struggle is compounded by ...
3
votes
1answer
95 views

Resources on 3D transforms, vectors, coordinate systems

Background: I'm helping engineers use software to create 3D geometry in a programmatic way (similar to OpenSCAD). The functions they need to call have inputs which are low-level geometry concepts: 3D ...
4
votes
2answers
169 views

Determinant applications for 16 year olds

I am teaching matrices, determinants and systems to a class of 16-year-olds. As you can expect, they do not know calculus or linear algebra. What they do know is a little bit of matrices (how to ...
3
votes
3answers
158 views

Defining/introducing vectors informally

A vector is a collection of elements whose order matters. If the elements represent comparable measurements it can be said that they together represent a direction and a magnitude (length). The ...
11
votes
2answers
213 views

How can one motivate the adjugate matrix?

The adjugate matrix of an $n \times n$ matrix $A$ is defined by $(\mathrm{adj}\ A)_{k\ell} = (-1)^{k+\ell}\,\det M(\ell,k)$, where $M(\ell,k)$ is the minor matrix obtained from $A$ by deleting row $\...
7
votes
3answers
164 views

Solving linear systems - Applications

I was thinking today applications of $\textbf{Ax}=\textbf{b}$ where $\textbf{A}\in\mathbb{R}^{m \times n}$. Specifically, I am interested to know what applications one might give to students, who don'...
16
votes
4answers
554 views

Applications and motivation of abstract linear algebra topics for engineers

This semester I'm teaching introductory linear algebra for engineering students, and I don't think I'm doing a good job explaining why these topics are important; specifically, everything having to do ...
15
votes
7answers
894 views

Why do we teach that every line is a linear function?

Teaching my precalculus class today, I noticed something very simple that I hadn't taken into account previously. The definition in our textbook read: "A linear function is a function defined by ...
13
votes
2answers
572 views

A simple explanation or derivation of Cramer's rule, suitable for secondary Algebra 2?

I am tutoring a high school student in Algebra 2. Her class has just covered systems of linear equations, mainly via the "substitution" and "elimination" methods. Her textbook concludes that unit ...
15
votes
6answers
472 views

How to get students in a under-graduate linear algebra course interested in determinants?

Before teaching the chapter on determinants in a linear-algebra course for beginning undergraduate students (mathematics and computer science, more specifically) I would like to give a small ...
10
votes
5answers
495 views

Textbook for first course in linear algebra

Since this does not appear to have been asked here before, I would like to solicit suggestions and recommendations, ideally with rationales, for a textbook in a first course in linear algebra. In my ...
22
votes
6answers
583 views

Too much motivation?

This is something that I felt like was difficult for me in some classes, especially lower division differential equations and linear algebra classes. I know professors want to motivate certain topics ...
9
votes
3answers
187 views

Teaching and motivating the use of Eigenvectors

I would like to know how to better demonstrate Eigenvectors. The texts that I have display the properties and methods to calculate them. There are plenty of great elementary examples to follow through ...
12
votes
3answers
334 views

Worksheet: Homology in Intro Lin Al

I am about to start grad school and I am trying to think seriously about teaching [you know, before I get swamped with my own coursework]. I wrote a hypothetical worksheet for an introductory linear ...
2
votes
2answers
232 views

Seeking your advice on books for abstract algebra and linear algebra

I am a college sophomore in the US with a major in mathematics and am an aspiring mathematician in the fields of computational complexity theory and cryptography. I would like to seek your advice and ...
14
votes
1answer
721 views

Linear algebra textbooks presenting an eclectic, geometric approach to the subject

I am teaching an undergraduate course in linear algebra this fall. I am dissatisfied with most existing textbooks, and indeed with the way in which this subject is usually taught. I hope to find a ...
14
votes
4answers
475 views

Key theorems in undergraduate linear algebra

I've been asked by an high school student what are the $5$ major theorems in Lang's Linear Algebra (and therefore, by extension, in an undergraduate linear algebra course). Firstly, I bluntly said ...
7
votes
1answer
482 views

How should I teach linear algebra and vector geometry together at high school?

I'm teaching mathematics at my former high school and the next topic will be vector geometry. When I attended high school, I was only taught vector geometry and never learnt anything about matrices ...
4
votes
3answers
2k views

Lang's Linear algebra or Introduction to linear algebra for an undergraduate

From a pedagogical as well as strictly mathematical perspective, which one of Lang's Linear algebra and Introduction to linear algebra would you recommend to an undergraduate with not much experience ...
5
votes
1answer
482 views

Self-study Linear Algebra textbook for ML and Stats

I am looking for a good linear/matrix algebra textbook, suitable for self-study, that covers topics relevant to statistics and machine learning. I have access to ...
7
votes
1answer
156 views

What are the good sources for Singapore Mathematics?

I am looking for the conceptual/visualisation way of tackling the algebra problems . I found that Singapore Maths caters this need. What are the good sources for Singapore Maths - any online classes ...
12
votes
2answers
640 views

Proving theorems on one's own: how long should one persist?

I've recently started learning linear algebra on my own. I always try to prove the theorems I encounter by myself, without looking at the book (only to check if my proof is correct), because I found ...
3
votes
0answers
37 views

Text book for 2nd Linear Algebra course [duplicate]

I stumbled across this site while searching for Hoffman and Kunze. There was a discussion about using HK for a beginning linear algebra course. I am teaching (for the first time) a 2nd course in ...
9
votes
11answers
1k views

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

How can we break down the complexity of eigenvalues/vectors to something that is more intuitive for students. I feel like the proofy way isn't a good intuitive representation of the mechanism that ...
12
votes
2answers
927 views

When is a good time to teach linear algebra?

When I was a student (in the 1970s) I was taught linear algebra as an "adjunct" to "engineering mathematics" such as differential equations. That was during my sophomore year, which seems a bit late, ...
10
votes
2answers
347 views

Linear algebra for engineers

When studying linear algebra in mathematics (I mean, for the people studying mathematics) there are many ways of approaching it, depending of your needs, however supposedly every mathematician should ...
10
votes
1answer
200 views

A more natural motivation for the appearance of generalized eigenvectors in linear system with repeated eigenvalue

When I teach constant coefficient linear differential equations, the usual guess of an exponential can be motivated because it is "approximately" a fixed point for the differentiation operator. The ...
6
votes
3answers
177 views

Should one visualize properties of a matrix or/and state its properties?

Sometimes, if you want students to manipulate some sort of special matrices (like, asking if they are invertible or ask for a LU decomposition, etc.), you have to possibility to state all properties ...
7
votes
4answers
969 views

Physics in Linear Algebra

Talking about physical phenomena related to a particular field of mathematics can be interesting for students and might further motivate their study of the subject. For instance, there are ...
15
votes
5answers
622 views

Should the cross-product in $\mathbb{R}^3$ be discussed in Linear Algebra?

I have not yet taught Linear Algebra, but I teach Computer Graphics regularly, which uses linear algebra at many junctures, and uses concepts such as the cross product. I have often been disappointed ...
19
votes
6answers
1k views

Is there a good way to explain determinants in an elementary linear algebra class?

Many colleges offer an an elementary linear algebra class for sophomore math, science, and economics majors. Such a class typically covers a chapter on determinants, including the following aspects: ...
21
votes
11answers
1k views

What is a good motivation/showcase for a student for the study of eigenvalues?

Courses about linear algebra make great demands on looking for eigenvalues and transforming matrices to diagonal matrices (or, at least, to Jordan normal form). This is somehow a technical, recipe-...
13
votes
5answers
287 views

Why teach back substitution with row reduction?

Many linear algebra books include two versions of row reduction for solving systems of linear equations: (1) Reduce to echelon form, and then use back substitution. (2) Reduce to reduced echelon ...