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 ...
  • 2,411
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, ...
  • 16.5k
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 ...
  • 16.5k
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 ...
  • 1,901
21 votes
Accepted

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-...
  • 16.5k
21 votes
Accepted

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 ...
  • 328
20 votes
Accepted

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 ...
  • 5,131
18 votes
Accepted

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 $n \times n$ ...
16 votes
Accepted

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 ...
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 ...
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 ...
  • 4,417
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., ...
  • 7,703
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 ...
11 votes
Accepted

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 ...
  • 4,417
11 votes

Lowercase vs. uppercase letters for matrix entries

$A_{jk}$ is sometimes used to mean the matrix $A$ with row $j$ and column $k$ deleted. [For example, see David Lay, Linear Algebra and its Applications, 4th edition, page 165.] To avoid confusion with ...
  • 18k
11 votes

Lowercase vs. uppercase letters for matrix entries

Sometimes a matrix name is suggestive: for example Jacobian or Ricci. We might use $\text{Jac}$ or $J$, or $\text{Ric}$ or $R$. In these situations it would be awkward to switch to lower case to ...
  • 8,083
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 ...
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 ...
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"... ...
10 votes
Accepted

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)} ...
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{...
  • 5,539
9 votes

Applications and motivation of abstract linear algebra topics for engineers

Two Four ideas: (1) "composing linear transformations": Use rotation, scaling, and shearing. If you extend to homogenous coordinates, you can include translations. Fundamental to all computer ...
9 votes

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

You might know (or not) enough computer science to know there are such things as functional programming languages. These are programming languages (the most popular are probably Scheme, ML, and ...
9 votes

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

That is a linear algebra course? So presumably before you get to this point of abstract vector space, you already did solution of systems of linear equations? For example, solution of matrix ...
  • 6,418
8 votes
Accepted

Worksheet: Homology in Intro Lin Al

I have a dim memory of a simple exercise I came up with as a TA. It was something like the vector space over the reals of polynomials in two variables, including x^2, y^2, and (x+y)^2. I then asked ...
8 votes

Worksheet: Homology in Intro Lin Al

Personally, I think this seems very appropriate. A few points: You might reconsider using a two character name for a vector. $xa$ looks like a multiplication. Perhaps just label the vertices $V_1$,...
8 votes

Linear algebra textbooks presenting an eclectic, geometric approach to the subject

I am intrigued by this book, but (a) I haven't used it myself, and (b) it lists @$200: Shifrin, Ted, and Malcolm Adams. Linear algebra: A geometric approach. Macmillan, 2ndEd, 2011. (Macmillan ...
8 votes

How to get students in a under-graduate linear algebra course interested in determinants?

To give a brief list of interesting applications: Volume obviously the lead application. It is not unreasonable to say determinants are volumes. Of course, they're more than that, their signed-...
8 votes
Accepted

How to come up with a Leslie matrix with convenient eigenvalues?

If I use your simplification that $f_0 = 0$, then I suggest just choosing a real eigenvalue $\lambda$ and writing out the relation for the other parameters: $$-\lambda^3+f_1s_0\lambda + f_2s_0s_1 = 0$...
  • 7,703

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