Questions tagged [linear-algebra]
For questions related to linear mappings, matrices, basis, determinants, eigenvalues and other topics commonly done in a linear algebra course.
82
questions
10
votes
5
answers
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 ...
- 1,238
3
votes
5
answers
366
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 ...
- 321
4
votes
2
answers
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.
...
- 28.7k
7
votes
7
answers
330
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 ...
- 309
16
votes
18
answers
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 ...
- 3,711
4
votes
1
answer
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 ...
user11702
10
votes
4
answers
451
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 ...
user11702
3
votes
1
answer
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 ...
- 3,711
6
votes
3
answers
352
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 ...
- 63
0
votes
2
answers
156
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
votes
2
answers
493
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
327
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 ...
- 71
11
votes
2
answers
405
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 ...
- 18.9k
9
votes
4
answers
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 "...
- 977
4
votes
2
answers
345
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 ...
- 141
0
votes
3
answers
202
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 ...
- 213
3
votes
3
answers
363
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 ...
- 125
40
votes
11
answers
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 ...
- 22.9k
2
votes
3
answers
251
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 ...
- 121
1
vote
0
answers
68
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. ...
- 183
4
votes
4
answers
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, ...
- 61
3
votes
4
answers
688
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 ...
- 28.7k
7
votes
1
answer
267
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
94
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 ...
- 121
7
votes
2
answers
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 ...
- 4,750
1
vote
0
answers
74
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 ...
- 103
0
votes
0
answers
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 ...
18
votes
8
answers
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 ...
- 1,004
-1
votes
1
answer
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 ...
- 13
6
votes
1
answer
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 ...
- 161
10
votes
3
answers
363
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 ...
- 111
6
votes
2
answers
233
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 ...
- 5,314
3
votes
2
answers
490
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....
- 231
10
votes
4
answers
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 ...
- 5,856
0
votes
1
answer
412
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 ...
- 13
6
votes
1
answer
164
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 $(a_1,a_2,\cdots,a_n)$...
- 4,119
8
votes
5
answers
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 ...
- 231
2
votes
3
answers
306
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 ...
- 4,750
3
votes
0
answers
512
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 ...
- 31
9
votes
2
answers
193
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 ...
- 191
5
votes
3
answers
193
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 ...
- 307
2
votes
1
answer
479
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 ...
- 29
10
votes
1
answer
325
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 ...
user507
7
votes
2
answers
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 ...
- 111
8
votes
4
answers
329
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 ...
- 273
6
votes
4
answers
662
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 ...
- 433
32
votes
12
answers
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 ...
- 4,119
11
votes
1
answer
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 ...
user9185
11
votes
1
answer
521
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 ...
- 1,207
3
votes
1
answer
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 ...
- 131