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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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7 votes
2 answers
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Producing defective matrix examples

It's easy to come up with examples of interesting diagonalizable matrices $A\in M_n(\mathbb{C})$ with a prescribed set of eigenvalues $\lambda_1,\dots,\lambda_n$ because you can start with your ...
Jeremy Brazas's user avatar
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
4 votes
3 answers
448 views

Regarding finding eigenvalues and minimal polynomial of an operator with limited tools while following Sheldon Axler

I am using the textbook Linear Algebra done right by Sheldon Axler (fourth edition) to teach an undergraduate linear algebra course. Please find here a link to the book here. Now Axler does not ...
user534666's user avatar
-4 votes
1 answer
107 views

Condense a logistic function around its midpoint [closed]

I have a function $$ f(x) = {1 \over 1 + e^{-k(x-0.5)}} $$ that plots a logistic curve that is symmetric around the point $0.5 / 0.5$. $k$ defines the steepness of the curve. I would now like to add ...
Fred's user avatar
  • 95
0 votes
1 answer
254 views

Help finding a YouTube channel for Linear Algebra

I remember watching a youtube playlist on a video about equation of a plane and about vectors in 3D. The video was fully animated in 3D and had good sound effects too. I feel like it had a clock ...
Broy Tennison's user avatar
6 votes
3 answers
199 views

Game project using linear algebra

Throughout this year, I have been in charge of a linear algebra course aimed at engineering school. In this course, I asked my students to work on a final project involving finding a winning strategy ...
Gabriel's user avatar
  • 61
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
4 votes
3 answers
284 views

Looking for web app resources for symbolic Gaussian elimination

I am looking for a web app software that takes step-by-step directions from a student to perform the linear combination operation on a matrix with symbolic coefficients (as opposed to just numbers). ...
Maesumi's user avatar
  • 1,410
0 votes
2 answers
75 views

How to intuitively connect Linear Equation in two variables and the graph of them? [closed]

I struggle with connecting graphs of linear equation with algebraic form like $x+y=p$. How do I develop the intuition that it represents a line that is sloping down and passes through value $p$ on $y$-...
Ashish Shukla's user avatar
5 votes
2 answers
481 views

What is the dimension of $\mathbb{R}$ over $\mathbb{Q}$?

Long time ago a student asked me what the dimension of $\mathbb{R}$ over $\mathbb{Q}$ is, and I said $$\dim_{\mathbb{Q}}\mathbb{R}=\mathfrak{c}$$ where $\mathfrak{c}$ is the cardinality of the ...
Peter Melech's user avatar
4 votes
2 answers
385 views

Programming and computation-focused textbook for introductory linear algebra

tl;dr I am looking for references which cover introductory abstract Linear Algebra but with a programming / computational approach. The only one I found is the Jupyter guide to Linear Algebra Long ...
Luiz Cordeiro's user avatar
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
2 votes
1 answer
187 views

Rediscovering euqation of line [closed]

I am studying (self learner) linear equations/equation of line and my idea is to discover the equations myself rather than try and understand ready-made equations available in text books. I am using X-...
Ashish Shukla'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
3 votes
5 answers
1k views

How do I teach the difference between Linear Equations and Equation of a Line

I am more of an Intuitive Learner and Teacher so while looking at Linear Equations chapter I see that they are teaching Equations of Line there, which in my opinion is wrong. See How I understand it ...
Ashish Shukla's user avatar
4 votes
2 answers
1k views

Is the Wronskian still assumed for graduate education?

About thirty years ago, in a practice GRE (Graduate Record Exam) math test in the US, a question assumed the student knew the definition of the Wronskian. I had never heard of this determinant before. ...
Joseph O'Rourke's user avatar
7 votes
7 answers
545 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
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
4 votes
1 answer
240 views

Tips for teaching mathematics with experiments?

I am teaching a first-linear-algebra course for mostly undergraduate Computer Science and Data Science students. Occasionally, I tried to do a bit computer experiments in class and ask students to ...
user avatar
10 votes
4 answers
522 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
3 votes
1 answer
104 views

Questions to help better understand the textbook

I am teaching a linear algebra class for math majors and non-majors out of the first 4 chapters of Lay's book. My plan is to have the students read a section prior to each class, have them answer a ...
David Steinberg's user avatar
6 votes
3 answers
433 views

Preparing to be a linear-algebra teacher: Any tips/suggestions?

I'm engineer and I love linear-algebra. I finished a couple of days the Linear-Algebra on OpenCourseMIT to refresh my memory and I have been doing ton of exercises. I would like to teach to engineer ...
user3321507's user avatar
0 votes
2 answers
168 views

Which context of the two contexts of the linear function concept should be taught first?

It appears in the current version of the Wikipedian article Linear function: In mathematics, the term linear function refers to two distinct but related notions:1 In calculus and related areas, a ...
humbleLearner's user avatar
4 votes
2 answers
537 views

Should I gave a make up lecture if some students found what I taught is a bit unclear?

This semester, I am teaching discrete math for computer science students. Today I taught solving linear recurrence equations. The way I did it was not rigors. Instead I used the method of advanced ...
user avatar
7 votes
1 answer
389 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
11 votes
2 answers
464 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
  • 21k
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
5 votes
2 answers
466 views

Applications of the annihilator from linear algebra

I am currently assisting a course for future teachers at university level for a joint education in maths and physics in Germany and I have a question regarding the use and possible application of the ...
Peter Strouvelle's user avatar
0 votes
3 answers
307 views

Highly intuitive yet comprehensive and easily readable (student friendly) book on linear algebra which do not focus much on applications, just basics

I came to know about Gilbert Strang's two books, "Introduction to Linear Algebra" and "Linear Algebra and its Applications". The first is the one used as the text in the 18.06 ...
Abhishek Ghosh's user avatar
4 votes
3 answers
533 views

What is the English word for the French "repère"?

I'm preparing a holiday class in computer graphics. The class will be held in English. I'm a French speaker and I'm fighting with some words which have lots of meanings to find the right one in the ...
mszmurlo's user avatar
  • 135
48 votes
14 answers
5k 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
2 votes
3 answers
300 views

Why do we typically only teach high-school students affine transformations of elementary functions?

A standard pre-calculus curriculum consists of the study of elementary functions: Polynomials, rational functions, (circular and hyperbolic) trigonometric functions, exponential functions, their ...
GradStudent's user avatar
1 vote
0 answers
75 views

History of business calculus/linear algebra curriculum

I will be teaching a combination of business calculus and business linear algebra, two classes that have been around awhile at my school. I’m assuming people are familiar with these types of classes. ...
Nights's user avatar
  • 183
6 votes
7 answers
494 views

Can we define length and perpendicularity not via an inner product?

A traditional way to model Euclidean geometry is to consider an inner product vector space $V$ and to define that the length of $v$ is $\sqrt{(v, v)}$ and that $v$ is perpendicular to $u$ iff $(v, u) =...
tekhnus's user avatar
  • 111
3 votes
4 answers
894 views

What is an algebraic explanation of why the product of the slopes of perpendicular lines is $-1$? [duplicate]

Q: What is a succinct, clear and purely algebraic explanation of why the product of the slopes of perpendicular lines is $-1$? Here I am aiming for high-school students (in the U.S.). I have a purely ...
Joseph O'Rourke's user avatar
7 votes
1 answer
306 views

The dimension theorem and pedagogy

The dimension theorem (the rank-nullity theorem) can be explained in many ways. I consider it as a consequence of the first isomorphism theorem/splitting lemma. When I teach undergrad matrix-theoretic ...
Matha Mota's user avatar
2 votes
0 answers
98 views

Teaching linear algebra, wacom tablet display of coordinate system, eigenvectors, markov chains

I am teaching linear algebra as part of an information retrieval course, which now occurs online. I have a Wacom tablet and free drawing software, sketchBook for artists, so can draw circles ellipses ...
ExpressionCoder'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
1 vote
0 answers
97 views

Exercises for explaning homothety, homothetic center, similarity on line and plane, free vector and vector space

I need the collection of exercises for such topics as: maps and transformations, composition of maps homothety, rotation homothety, homothetic center similarities of the line and the plane free ...
paus's user avatar
  • 83
0 votes
0 answers
157 views

How can I introduce the idea of eigenvectors and matrix decompositions to a general audience in an engaging manner?

So I'm doing a freelance writing job, writing a script for a YouTube video about eigenvectors/values. It took me a while to decide what the focus was going to be, but I finally settled on focusing on ...
Mikayla Eckel Cifrese'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
  • 1,411
-1 votes
1 answer
116 views

How to start tackling a 200 page big script? [closed]

I am really not sure how to tackle a 200 page huge script with a lot of information about the topic linear algebra. Writing down the definitions would not make sense in my opinion, because if I want ...
SelfLearner's user avatar
6 votes
1 answer
199 views

Notation in the definition of matrix multiplication

When matrix multiplication is introduced, it is usually introduced with an additional variable: Given two multiplicable matrices $A$, $B$, one defines the product $C=AB$ to be the matrix given by some ...
blue's user avatar
  • 161
10 votes
3 answers
436 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
6 votes
2 answers
270 views

How to come up with a Leslie matrix with convenient eigenvalues?

A three by three Leslie matrix looks like $$ \begin{bmatrix} f_0 & f_1 & f_2 \\ s_0 & 0 & 0 \\ 0 & s_1 & 0 \end{bmatrix}, $$ where $f_0 \ge 0$ and everything else is ...
Tommi's user avatar
  • 7,378
4 votes
2 answers
782 views

Selected Exercise for Linear Algebra Done Right Edition 3 [closed]

I am self learning the book Linear Algebra Done Right. I tried to complete all exercises in each chapter. I am currently at Chapter 3 and found that it is not feasible to complete all of the exercises....
JOHN's user avatar
  • 251
12 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
  • 5,986
0 votes
1 answer
501 views

Gil Strang or Peter Lax? Which linear algebra book to use? [closed]

I have heard that Peter Lax's linear algebra book is the hardest you can find on the subject, but that it is the greatest linear algebra text in the world. But I also hear great things about Gil ...
user12029's user avatar
6 votes
1 answer
170 views

Difficulty in teaching the coordinates of a vector with respect to a basis $\{v_1,v_2,\ldots,v_n\}$

Let $V$ be a finite dimensional vector space and let $B=\{v_1,v_2,\cdots,v_n\}$ be a basis of $V$. If a vector $v$ can be written as $$v=a_1v_1+a_2v_2+\cdots+a_nv_n,$$ we call $(a_1,a_2,\cdots,a_n)$...
Zuriel's user avatar
  • 4,275
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