Questions tagged [linear-algebra]
For questions related to linear mappings, matrices, basis, determinants, eigenvalues and other topics commonly done in a linear algebra course.
99 questions
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Recommendations on the number of exercises to do from Linear Algebra and Its Applications 6th edition (by Lay & McDonald)
So I wanted to review my intro to Linear Algebra course in prep for next semester classes (starts Aug 2025/Fall since taking 2 gap semesters starting Dec 12th for personal reasons) like "a second ...
- 193
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votes
3
answers
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Finding a Linear Algebra reading topic
I'm looking for a linear algebra topic to assign to a group of undergraduate engineering students who are not majoring in mathematics. The assignment should meet the following criteria:
The topic ...
- 91
3
votes
3
answers
310
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Question on implementation of PBL (problem or project-based learning) in Linear Algebra course
I am trying to implement some version of a Project or Problem-Based Learning methodology (which I will just refer to as "PBL"[1]) in Linear Algebra and I am not sure how classes are ...
- 769
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4
answers
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Questions about writing a Linear Algebra textbook, with Earth Science applications
I am writing a Linear Algebra textbook that is supposed to have several Earth Science applications (Geophysics, Atmospheric Science) given that I come from an Earth/Atmospheric Science background. I ...
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2
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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 ...
- 221
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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 ...
- 474
4
votes
3
answers
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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 ...
- 141
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votes
1
answer
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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 ...
- 95
0
votes
1
answer
292
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 ...
6
votes
3
answers
254
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 ...
- 61
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votes
14
answers
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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, ...
- 749
4
votes
3
answers
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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). ...
- 1,410
0
votes
2
answers
93
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 $y$-...
- 391
5
votes
2
answers
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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 ...
- 153
4
votes
2
answers
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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 ...
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votes
4
answers
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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 ...
- 404
2
votes
1
answer
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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-...
- 391
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votes
5
answers
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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 ...
- 2,334
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 ...
- 391
4
votes
2
answers
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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.
...
- 30.2k
7
votes
7
answers
682
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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 ...
- 319
19
votes
18
answers
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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 ...
- 3,890
4
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1
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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 ...
user11702
10
votes
4
answers
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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 ...
user11702
3
votes
1
answer
107
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 ...
- 3,890
6
votes
3
answers
452
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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 ...
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2
answers
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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 ...
4
votes
2
answers
544
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 ...
user11702
7
votes
1
answer
420
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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 ...
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votes
2
answers
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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 ...
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12
votes
4
answers
4k
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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 "...
- 1,017
5
votes
2
answers
500
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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 ...
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3
answers
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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 ...
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3
answers
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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 ...
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votes
16
answers
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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 ...
- 26k
2
votes
3
answers
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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 ...
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0
answers
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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. ...
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6
votes
7
answers
528
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) =...
- 111
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votes
4
answers
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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 ...
- 30.2k
7
votes
1
answer
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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 ...
- 173
2
votes
0
answers
101
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 ...
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votes
2
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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 ...
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1
vote
0
answers
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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 ...
- 83
0
votes
0
answers
159
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 ...
19
votes
8
answers
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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 ...
- 1,431
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votes
1
answer
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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 ...
- 13
6
votes
1
answer
203
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 ...
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votes
3
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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 ...
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votes
2
answers
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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 ...
- 8,093
4
votes
2
answers
839
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....
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