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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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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 ...
• 475
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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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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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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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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 ...
• 11.6k
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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 ...
• 11.6k
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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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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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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 ...
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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 ...

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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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 ...
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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 ...
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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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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 ...
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