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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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Find unit price of each item given the total spent [on hold]

I am trying to help my daughter in her math and there is this question I can't quite get my head around, The sum is: Three friends go into a book shop. Salma buys a cook book and a novel, she pays \$...
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3answers
223 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 ...
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0answers
91 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 ...
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0answers
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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 ...
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3answers
152 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 ...
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1answer
122 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 ...
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1answer
199 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 ...
6
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1answer
102 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 ...
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4answers
328 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 ...
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9answers
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 ...
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1answer
268 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 ...
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1answer
325 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 ...
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1answer
95 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 ...
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2answers
172 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 ...
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3answers
160 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 ...
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2answers
218 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 $\...
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3answers
165 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'...
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4answers
579 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
906 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 ...
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2answers
657 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 ...
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6answers
475 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 ...
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5answers
498 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 ...
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6answers
590 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 ...
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3answers
191 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 ...
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3answers
340 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 ...
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2answers
239 views

Seeking your advice on books for abstract algebra and linear algebra

I am a college sophomore in the US with a major in mathematics and am an aspiring mathematician in the fields of computational complexity theory and cryptography. I would like to seek your advice and ...
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1answer
750 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 ...
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4answers
481 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 ...
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1answer
505 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 ...
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3answers
2k 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 ...
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1answer
502 views

Self-study Linear Algebra textbook for ML and Stats

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 ...
7
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1answer
160 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 ...
12
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2answers
688 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 ...
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0answers
38 views

Text book for 2nd Linear Algebra course [duplicate]

I stumbled across this site while searching for Hoffman and Kunze. There was a discussion about using HK for a beginning linear algebra course. I am teaching (for the first time) a 2nd course in ...
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11answers
1k 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 ...
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2answers
962 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, ...
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2answers
355 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 ...
10
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1answer
202 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 ...
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3answers
178 views

Should one visualize properties of a matrix or/and state its properties?

Sometimes, if you want students to manipulate some sort of special matrices (like, asking if they are invertible or ask for a LU decomposition, etc.), you have to possibility to state all properties ...
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4answers
1k 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 ...
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5answers
631 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 ...
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6answers
1k 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: ...
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11answers
2k 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-...
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5answers
292 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 ...