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41 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
  • 2,568
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
  • 475
24 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
  • 17.4k
22 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
  • 17.4k
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
22 votes
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What Basic Math Skills Should Be Expected of Students in a University-Level Linear Algebra Course?

What basic math skills (e.g., polynomial manipulation, trigonometry, complex numbers) is reasonable for us to expect students to have when they enroll in a linear algebra course at the university ...
Justin Skycak's user avatar
21 votes
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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
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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 ...
DavidButlerUofA'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

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
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
  • 9,699
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
  • 7,607
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
  • 4,753
13 votes

What Basic Math Skills Should Be Expected of Students in a University-Level Linear Algebra Course?

For what it's worth, when I cover eigenvalues of a $3\times 3$ matrix, I only assign matrices where either zero is an eigenvalue, or there is a helpful factor like $(1-\lambda)$ that stays on the ...
user1149748's user avatar
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

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
  • 221
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
  • 11k
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
  • 21k
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

Big list of "interesting" abstract vector spaces

The set of solutions to a system of linear homogeneous ODEs is a vector space, and the dimension of this vector space is equal to the total order of the system. The idea that every solution is the ...
Michael Seifert's user avatar
10 votes

Big list of "interesting" abstract vector spaces

Let $\Omega$ be a set, and let $\mathcal A$ be an algebra of subsets of $\Omega$. Then $\mathcal A$ is a vector space over the field $\mathbb F_2 = \{0,1\}$, with the operation $$ E \mathbin{\Delta} ...
Gerald Edgar's user avatar
  • 7,607
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
  • 5,629
9 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
  • 5,832
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

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
  • 11k
9 votes
Accepted

What is the dimension of $\mathbb{R}$ over $\mathbb{Q}$?

I remember being confused by this as an undergraduate, or to be more honest, blind to the fact that I held a fundamental misconception that would take years to realise. A student in a first linear ...
user52817's user avatar
  • 11k
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$...
Nick C's user avatar
  • 9,699

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