40 votes

Should college mathematics always be taught in such a way that real world applications are always included?

I have worked with a lot of students coming out of courses such as yours who: passed the course by blindly memorising proofs, theorems, and algorithms; learnt nothing (lasting) except solving some ...
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  • 2,366
25 votes

Why do we teach that every line is a linear function?

The usage you object to is, in fact, the original meaning of "linear". "Linear" means "having to do with lines". The notion of "linear" in the sense of "linear transformation" is a more modern, ...
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  • 16.1k
22 votes
Accepted

Is there a good way to explain determinants in an elementary linear algebra class?

I subscribe to the down with determinants school of thought. In an elementary linear algebra course, particularly for non-mathematicians, then I think that determinants are as useful as your appendix[...
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21 votes
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What is a good motivation/showcase for a student for the study of eigenvalues?

Obtaining formulas for the $n$-th term in a linear recurrence, such as Fibonacci numbers, is one application that certainly does not overtly mention linear algebra in the set-up.
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  • 13.4k
21 votes

Should college mathematics always be taught in such a way that real world applications are always included?

At my University, there are four different first-semester Linear Algebra courses taken by Undergraduates: Math 214, Applied Linear Algebra, is "an introduction to matrices and linear algebra... The ...
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  • 16.1k
21 votes

Why do some linear algebra courses focus on matrices rather than linear maps?

Welcome Kostya! The mapping view is definitely important, but I don't think it's supreme. For me here's how I think about it. There are three ways to think about (basic) linear algebra: As a theory of ...
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  • 1,873
21 votes
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Concrete vectors spaces without an obvious basis or many "obvious" bases?

Some physical examples from physics: Consider two spaceships that meet each other in deep space with arbitrary orientations (pitch, roll, and yaw). Even if they take the origin to be the midpoint ...
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  • 326
20 votes
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Is Linear Algebra Done Right too much for a beginner?

Unguided self-study of mathematics is difficult, and harder for someone with little experience at it. It is normal to take time to advance. One should think in terms of months not hours. A typical one ...
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  • 4,958
20 votes
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Big list of "interesting" abstract vector spaces

Here are some more examples: $C[a,b]$, the set of continuous real-valued functions on an interval $[a,b]$. This abstract vector space has some very nice properties that make it very good for a first-...
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  • 16.1k
18 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

I like Markov chains and Google PageRank (which is essentially a special kind of Markov chain). It doesn't take very long to explain and motivate Markov chains and to argue that the probability ...
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18 votes
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How to get students in a under-graduate linear algebra course interested in determinants?

I have found it motivates to explain the determinant as computing a volume. One can work through and convince for $2 \times 2$ and $3 \times 3$ matrices, and perhaps only hint at the $n \times n$ ...
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17 votes
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Should the cross-product in $\mathbb{R}^3$ be discussed in Linear Algebra?

The problem is that there are multiple main thrusts of linear algebra. For many students (especially pure math majors), linear algebra is their first introduction to abstract algebra. When a course ...
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  • 2,671
16 votes
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Helping a student exasperated by abstract concepts in linear algebra

Definitions and other facts One thing I find particularly helpful with Linear Algebra is to help the student deal with the definitions in multiple ways. In Linear Algebra there are definitions, and ...
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16 votes

Should college mathematics always be taught in such a way that real world applications are always included?

I believe you need to listen beyond what your student is saying. Your student is not saying "I want to do some applications in class." What your student is really saying is "I'm bored and lost and ...
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15 votes
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Why teach back substitution with row reduction?

One reason to know back substitution is that it is relevant when doing numerical mathematics. A standard procedure to numerically solve linear systems $Ax =b$ especially if one wants to solve for ...
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  • 7,572
14 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

Imagine a linear mapping $f: R^2 \to R^2, e_1 \mapsto (1.5, 0.5), e_2 \mapsto (0.5, 1.5)$. (As long as $R$ contains the numbers $1.5$ and $0.5$, it could be any ring. The real numbers serve as the ...
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  • 4,733
13 votes

Teaching LU Factorization in a sophomore-level Linear Algebra course

Poole's Linear Algebra: A Modern Introduction, 2nd edition, relegates the non-square case of the LU factorization to an exercise. Strang's Introduction to Linear Algebra, 5th edition, does square ...
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  • 4,308
13 votes

Concrete vectors spaces without an obvious basis or many "obvious" bases?

Two more examples: The set of infinite Fibonacci-type sequences (those of the form $a_n=a_{n-1} + a_{n-2}$) (with point-wise addition and scaling) forms a 2-dimensional (real) vector space. E.g., ...
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  • 7,686
12 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

I have always found the standard motivations for eigenvalues to be a little artificial. The primary application for eigenvalues is ultimately diagonalization and there are several ways you could try ...
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  • 563
12 votes

Should college mathematics always be taught in such a way that real world applications are always included?

I challenge the assertion that students need to see applications in everything. When I first started teaching I labored under the delusion that I should explain connections to physics whenever I ...
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11 votes

Why teach back substitution with row reduction?

When I teach the $L U$-decomposition, mentioned in quid's answer, I use the following mnemonic: Single $\vec{b}$? Use GE. Several to do? Use LU. The point being that if you need to solve $A \...
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11 votes
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Proving theorems on one's own: how long should one persist?

I think it's very commendable to try proving things yourself first; even a failed attempt has value. However, it's also important to learn from others' proofs, so don't be afraid to sneak a peek at ...
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  • 4,308
10 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

For a real showcase, I recommend a scenario where resonance frequencies play a role. Suspension bridges are real-world objects which are delicate enough that soldiers are usually not allowed to march ...
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  • 1,791
10 votes
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When is a good time to teach linear algebra?

My personal opinion. Anytime is a good time to teach linear algebra. I'd like to see at least two courses in the curriculum. Ok, more accurately: elementary matrix theory: solving systems, matrix ...
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10 votes

Too much motivation?

Let me propose a non-standard distinction between two terms (in the context of teaching): An application is a problem or a task outside the main scope of the course with a solution presented using ...
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10 votes

Too much motivation?

Yes, I agree that there is too much motivation... in the sense of sloppy presentations where it is very unclear to students exactly what is what. I think a lot of the time the refrain "is this on the ...
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10 votes

Applications and motivation of abstract linear algebra topics for engineers

Of course it depends on how much time you're willing to spend on this. If the answer is "very little" then no chance that you can say something more than "in the future this will be useful for you"... ...
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10 votes
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How can one motivate the adjugate matrix?

Here is one way to put the rabbit in the hat before pulling it out: Derive the general formula for the inverse of an invertible matrix. It ends up having the form $A^{-1} = \frac{1}{\textrm{Det}(A)} ...
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10 votes

Concrete vectors spaces without an obvious basis or many "obvious" bases?

I think you're on the right track with the polynomials. They're not wrong that $(a,b,c)\mapsto (x\mapsto ax^2+bx+c)$ is an obvious linear isomorphism from $\mathbb R^3$ to what I will call $\text{...
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  • 5,539
9 votes
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What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

Here's an example I use for myself. I don't teach this topic in regular class but I have used this example in private conversations with advanced students. Think of an object (perhaps a globe) that ...
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  • 2,496

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