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 ...
- 2,538
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 ...
- 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, ...
- 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 ...
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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 ...
- 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-...
- 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 ...
- 6,171
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 ...
- 5,728
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
$n \times n$ ...
- 29.5k
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 ...
- 9,053
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 ...
- 161
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 ...
- 24.3k
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., ...
- 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 ...
- 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 ...
- 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 ...
- 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 ...
- 10.7k
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 ...
- 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 ...
- 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 ...
- 24.3k
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"...
...
- 1,715
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)} ...
- 24.3k
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{...
- 5,599
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 ...
- 29.5k
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 ...
- 2,484
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 \...
- 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 ...
- 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-...
- 10.7k
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)...
- 5,762
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