# 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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### 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 ...
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 ...
3k 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 ...
165 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-...
1k 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 ...
400 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 ...
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. ...
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 ...
470 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 ...
346 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 ...
230 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 ...
420 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 ...
357 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 ...
101 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 ...
2k 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 ...
158 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 ...
495 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 ...
330 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 ...
2k 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 "...
359 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 ...
3k 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 ...
210 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 ...
378 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 ...
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-...
255 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 ...
1 vote
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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. ...
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 ...
283 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, ...
371 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 ...
336 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 ...
670 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 ...
701 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 ...
268 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 ...
212 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 ...
95 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 ...
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 ...
1 vote
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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 ...
585 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 ...
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 ...
152 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 ...
114 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 ...
186 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 ...
5k 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 ...
239 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 ...
507 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....
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 ...
194 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 ...
418 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 ...
### 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 $(a_1,a_2,\cdots,a_n)$...