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 ...
Wrzlprmft's user avatar
  • 2,538
27 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 ...
Mark H's user avatar
  • 465
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, ...
mweiss's user avatar
  • 17.3k
22 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 ...
Nate Bade's user avatar
  • 1,941
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 ...
mweiss's user avatar
  • 17.3k
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-...
mweiss's user avatar
  • 17.3k
21 votes

What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations?

I am really at a loss as to why this is happening and what should be the correct remedy, as it seems that it is not easy to address this late into their study. Also there seems to be no improvement ...
Justin Skycak's user avatar
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 ...
Dan Fox's user avatar
  • 5,728
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 $n \times n$ ...
Joseph O'Rourke's user avatar
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 ...
DavidButlerUofA's user avatar
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 ...
Greg Blumberg's user avatar
16 votes

Why do we teach linear algebra in precalculus classes?

Vector algebra is a standard 3rd-semester calculus topic (e.g., see OpenStax Calculus 3, Ch. 2-3). This includes calculations of the dot product, cross product, and related values. Standard ...
Daniel R. Collins's user avatar
15 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., ...
Nick C's user avatar
  • 9,184
14 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 ...
Gerald Edgar's user avatar
  • 7,369
14 votes

What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations?

I'd like to offer a bit of a frame challenge here. Whenever I made these sorts of mistakes during my mathematics education it was normally because I didn't understand what the symbols in the more ...
andypea's user avatar
  • 257
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 ...
J W's user avatar
  • 4,625
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 ...
James S. Cook's user avatar
12 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 ...
user52817's user avatar
  • 10.3k
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 ...
Sue VanHattum's user avatar
  • 20.1k
11 votes

What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations?

I deal with a dash of this phenomenon in the context of teaching an "advanced programing techniques" course (i.e., 2nd semester programming) in a community-college CS program. There's ...
Daniel R. Collins's user avatar
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"... ...
Nicola Ciccoli's user avatar
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)} ...
Steven Gubkin's user avatar
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{...
Matthew Daly's user avatar
  • 5,599
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 ...
Joseph O'Rourke's user avatar
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 ...
Alexander Woo's user avatar
9 votes

Big list of "interesting" abstract vector spaces

The vector space $V = C^{\infty}(\mathbb{R},\mathbb{R})/\mathbb{R}[x]$ of smooth functions modulo polynomials. Note that $ d/dx \colon V\to V $ is an isomorphism, so that we have a nice inverse $\int \...
Gaussler's user avatar
  • 191
9 votes

Why do we teach linear algebra in precalculus classes?

The College Board made curriculum decisions for their new AP Precalculus course that align with sentiments you express. The course is divided into four units, where unit four is titled Functions ...
user52817's user avatar
  • 10.3k
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-...
James S. Cook's user avatar
8 votes

Notation for change of basis matrix

I like to use $ {}_{\mathcal{C}}A_{\mathcal{B}}$ for the change from $\mathcal{B}$ to $\mathcal{C}$ because then the subscripts match up when you try to compose the matrices (in the usual convention)...
Jessica B's user avatar
  • 5,762

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