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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2answers
121 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 ...
4
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2answers
473 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 ...
7
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1answer
292 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 ...
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2answers
322 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 ...
9
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4answers
439 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 "...
4
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2answers
252 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 ...
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3answers
174 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 ...
3
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3answers
196 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 ...
33
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11answers
2k 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 ...
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3answers
221 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 ...
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0answers
59 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. ...
4
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4answers
257 views

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

A natural way to reason about Euclidean geometry using modern mathematical language is to define Euclidean space as an affine space $A$ directed by a finite-dimensional real vector space $V$. However, ...
3
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4answers
618 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 ...
7
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1answer
236 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 ...
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0answers
82 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 ...
7
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2answers
1k 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 ...
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0answers
46 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 ...
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0answers
136 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 ...
18
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8answers
3k 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 ...
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1answer
113 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 ...
5
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1answer
165 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 ...
10
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3answers
276 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 ...
6
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2answers
203 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 ...
3
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2answers
319 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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3answers
1k 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 ...
0
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1answer
353 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 ...
6
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1answer
151 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 $(a_1,a_2,\cdots,a_n)$...
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5answers
4k 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 ...
1
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3answers
281 views

Word for the dimension of the vector space in which a vector lives?

The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $\mathbb{R}^3$. I am tempted to say that it has "length $3$". But then ...
3
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0answers
349 views

A proof based Multivariable Calculus and Linear Algebra

May I know how can I teach a proof-based Multivariable Calculus and linear algebra as a single course? While there are quite a few known books in the field such as: 1) Vector Calculus, Linear Algebra ...
9
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2answers
181 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
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3answers
181 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
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1answer
461 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
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1answer
287 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 ...
7
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2answers
186 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 ...
8
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4answers
298 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 ...
6
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4answers
575 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 ...
32
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12answers
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 ...
11
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1answer
1k 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
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1answer
447 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
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1answer
99 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
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2answers
216 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
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3answers
180 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 ...
10
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2answers
297 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
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3answers
189 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'...
18
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4answers
875 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 ...
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7answers
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 ...
13
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2answers
2k 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 ...
16
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6answers
609 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
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5answers
590 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 ...