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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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 ...
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1
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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 ...
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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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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 ...
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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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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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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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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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1
answer
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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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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 ...
user507
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2
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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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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 ...
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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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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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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 ...
user9185
11
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1
answer
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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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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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2
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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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3
answers
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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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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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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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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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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 ...
- 1,796
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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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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 ...
- 7,752
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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 ...
- 2,611
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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 ...
user5108
9
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3
answers
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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 ...
- 191
14
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3
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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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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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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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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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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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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 ...
- 1,131
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Self-study Linear Algebra textbook for Machine Learning and Statistics
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 ...
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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 ...
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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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4
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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 ...
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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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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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2
answers
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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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11
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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 ...
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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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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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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 ...
user230
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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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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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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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3
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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 ...
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