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Questions tagged [linear-algebra]

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

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48 votes
14 answers
4k views

Big list of "interesting" abstract vector spaces

When introducing an abstraction it is important (in my opinion) to have a wide variety of examples of this abstraction. Since finite dimensional real vector spaces are classified up to isomorphism by ...
Steven Gubkin's user avatar
34 votes
12 answers
7k 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 ...
Zuriel's user avatar
  • 4,275
26 votes
13 answers
3k 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-...
Markus Klein's user avatar
  • 9,438
26 votes
6 answers
1k 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 ...
user avatar
25 votes
14 answers
17k views

What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations?

I have helped to TA and taught several courses with mixtures of advanced undergraduate and early graduate students in engineering/STEM. These courses are the classics: signal processing, control, ...
Fraïssé's user avatar
  • 749
23 votes
6 answers
2k 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: ...
Jim Belk's user avatar
  • 8,279
21 votes
4 answers
979 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 ...
Javier's user avatar
  • 675
19 votes
18 answers
3k views

Concrete vectors spaces without an obvious basis or many "obvious" bases?

I am teaching a class on linear algebra to sophomore and junior science majors, and am having some trouble illustrating the difference between $\mathbb{R}^n$ and an n-dimensional vector space. The ...
David Steinberg's user avatar
19 votes
8 answers
4k views

Why do some linear algebra courses focus on matrices rather than linear maps?

I hope the essence of the question is clear from the title. There are obvious advantages to making the linear map the central notion of a linear algebra course: the notion can be illustrated with ...
Kostya_I's user avatar
  • 1,401
19 votes
4 answers
4k views

Why do we teach linear algebra in precalculus classes?

When I took precalculus, we learned about polynomials and how to factor them, we learned about trigonometry and lots of great and useful identities there, and we learned about matrices. They didn't ...
Joel Croteau's user avatar
19 votes
1 answer
2k 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 ...
Frank Thorne's user avatar
  • 2,249
17 votes
7 answers
1k 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 ...
Andrew's user avatar
  • 1,786
16 votes
6 answers
703 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 ...
quid's user avatar
  • 7,712
15 votes
5 answers
1k 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 ...
Joseph O'Rourke's user avatar
15 votes
1 answer
2k 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 ...
Adrian's user avatar
  • 253
14 votes
7 answers
8k views

What Basic Math Skills Should Be Expected of Students in a University-Level Linear Algebra Course?

I am currently teaching a university-level linear algebra course and recently encountered an issue that made me question the assumed foundational math skills for students in this course. The issue ...
LeafGlowPath's user avatar
14 votes
3 answers
666 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 ...
Jim Belk's user avatar
  • 8,279
14 votes
3 answers
475 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 ...
Eric Stucky's user avatar
14 votes
2 answers
2k 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, ...
Tom Au's user avatar
  • 1,502
14 votes
4 answers
1k 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 ...
Dal's user avatar
  • 1,111
13 votes
14 answers
3k 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 ...
David BasedMathematician Coven's user avatar
13 votes
2 answers
3k 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 ...
mweiss's user avatar
  • 17.4k
12 votes
5 answers
747 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 ...
Michael Joyce's user avatar
11 votes
4 answers
4k views

Computing eigenvalues by hand without determinants

I'm teaching a linear algebra class and I'm considering presenting eigenvectors and eigenvalues without using determinants, as in Axler's book Linear Algebra Done Right. (See also Axler's paper "...
littleO's user avatar
  • 1,007
11 votes
4 answers
4k views

Notation for change of basis matrix

As far as I can tell, it's only a slight exaggeration to say that every text has a different notation for a change of basis matrix from (say) $\mathcal{B}$ to $\mathcal{C}$. That's not even to talk ...
kcrisman's user avatar
  • 5,976
11 votes
2 answers
459 views

Teaching LU Factorization in a sophomore-level Linear Algebra course

I teach this course from David Lay's Linear Algebra and Its Applications, which on the whole is a great textbook and explains things well. It does not explain the steps of LU factorization well, so I ...
Sue VanHattum's user avatar
  • 20.8k
11 votes
1 answer
546 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 ...
AegisCruiser's user avatar
  • 1,427
11 votes
1 answer
268 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 ...
Jon Bannon's user avatar
  • 6,173
11 votes
1 answer
2k 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 ...
user avatar
11 votes
2 answers
599 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 ...
Ana Galois's user avatar
10 votes
5 answers
2k views

Lowercase vs. uppercase letters for matrix entries

For a matrix $A$ in, say for instance, $\mathbb{R}^{m \times n}$, there are at least two different conventions to denote its entry at position $(j,k)$: Denote the entry as $a_{jk}$. Denote the entry ...
Jochen Glueck's user avatar
10 votes
2 answers
347 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 $\...
Mark Wildon's user avatar
10 votes
4 answers
519 views

Should students in a first university linear algebra class be taught to write simple proofs?

I am teaching an introductory linear algebra courses for undergraduate students in math, computer science, or data science at a liberal art university. Most of the students have not decided their ...
user avatar
10 votes
3 answers
429 views

Can anyone recommend good software for verifying row reduction steps?

I work as a tutor for linear algebra. Students will often have a problem that requires them to row reduce, and they'll make an arithmetic mistake somewhere. Often they'll ask for my help finding ...
user12682's user avatar
  • 111
10 votes
2 answers
632 views

Textbook for 2nd linear algebra course

I am teaching (for the first time) a 2nd course in linear algebra. The students will have had a beginning course in linear algebra and a beginning course in abstract algebra. I am considering Hoffman ...
Angela's user avatar
  • 139
10 votes
1 answer
353 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 ...
user avatar
9 votes
5 answers
6k views

Is Linear Algebra Done Right too much for a beginner?

I have asked in Mathematics stackexchange, but I think asking here is more appropriate. I am self studying the book Linear Algebra Done Right by Axler. That's how I started using the great Stack ...
JOHN's user avatar
  • 251
9 votes
3 answers
240 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 ...
Vass's user avatar
  • 191
8 votes
4 answers
383 views

Make a matrix algebra course (1st university year) more "project-based"

Among other courses, I'm teaching a (basic) matrix algebra course for 1st year university students (they are studying Economics, and the cursus leads them to management, finance, or econometrics in ...
Basj's user avatar
  • 273
8 votes
2 answers
238 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 ...
Sheri A.'s user avatar
  • 121
8 votes
3 answers
3k views

Self-study Linear Algebra textbook for Machine Learning and Statistics

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 ...
neuron's user avatar
  • 221
8 votes
3 answers
6k 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 ...
Dal's user avatar
  • 1,111
8 votes
2 answers
205 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 ...
PhysicsTeacher's user avatar
7 votes
4 answers
3k 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 ...
user avatar
7 votes
7 answers
515 views

Creative problems in 2D vector geometry

What are some "interesting" and creative problems or exercises on specifically 2-dimensional vector geometry that a high school student might find compelling to solve? The class' current ...
Damian Reding's user avatar
7 votes
3 answers
246 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'...
user153764's user avatar
7 votes
2 answers
2k views

Why does a first course in linear algebra teach QR-decomposition?

I am teaching a "linear algebra for engineers" course, and am currently building my lectures on Gram-Schmidt, QR-decomposition and least squares equation solving. $\bullet$ I can motivate ...
David E Speyer's user avatar
7 votes
1 answer
214 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 ...
dexterous's user avatar
  • 171
7 votes
1 answer
382 views

Vector geometry as a prelude to linear algebra

When I was in grade 11, I was fortunate enough to attend a high school that offered an optional course in vector geometry. The course was taught out of the book Analytic Geometry with an Introduction ...
Dave's user avatar
  • 71
7 votes
1 answer
653 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 ...
Huy's user avatar
  • 699