39
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 calculating exercises, a very vague idea of some terminology;
had no idea what they just learnt, why they learnt it, and how it relates too their field of study; ...
25
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, restricted notion.
21
I subscribe to the down with determinants school of thought. In an elementary linear algebra course, particularly for non-mathematicians, then I think that determinants are as useful as your appendix[1]. Nevertheless, they are on the syllabus so I have to teach them.
Here's what I do. I tell the students the truth: that determinants used to be a key part ...
21
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 matrices, linear maps and systems of linear equations. It is a fascinating result that when you formalize these three views the mathematical objects are ...
20
Obtaining formulas for the $n$-th term in a linear recurrence, such as Fibonacci numbers, is one application that certainly does not overtly mention linear algebra in the set-up.
20
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 emphasis is on concepts and problem solving. The sequence 214-215 is not for math majors. It is designed as an alternate to the sequence 215-216 for engineering ...
20
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 semester class in linear algebra does not cover every detail in the typical linear algebra textbook, nor require a student to work every exercise in that book. ...
18
I like Markov chains and Google PageRank (which is essentially a special kind of Markov chain). It doesn't take very long to explain and motivate Markov chains and to argue that the probability distribution at time $n$ is the $n$'th power of the transition matrix times the distribution at time $0$. You can then start talking about how to calculate powers ...
18
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
$n \times n$ generalization, when $|\det(M)|$ is the volume of the $n$-dimensional
parallelepiped spanned by the column vectors of $M$.
&...
17
The problem is that there are multiple main thrusts of linear algebra.
For many students (especially pure math majors), linear algebra is their first introduction to abstract algebra. When a course is designed with this in mind, rigor and formalism takes higher precedence as we want as much of the material to be reusable in the context of modules over ...
16
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-semester linear algebra course:
a. It has a natural inner product on it, given by $\langle f, g \rangle = \int_a^b f(t)g(t) \, dt$
b. It contains the (infinite-...
15
One reason to know back substitution is that it is relevant when doing numerical mathematics.
A standard procedure to numerically solve linear systems $Ax =b$ especially if one wants to solve for more then one $b$ (which is very common) is to perform an $LU$-decomposition of $A$ (or something similar like a Cholesky decomposition), that is one decomposes $A$...
15
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 there are properties that things have that are always true but aren't definitions. (They could have been chosen as definitions, but the chosen definition is more ...
15
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 this is rapidly becoming a waste of time for me, so I'm making this suggestion because I care enough about my own learning and trying to connect with you is how I ...
14
Imagine a linear mapping $f: R^2 \to R^2, e_1 \mapsto (1.5, 0.5), e_2 \mapsto (0.5, 1.5)$. (As long as $R$ contains the numbers $1.5$ and $0.5$, it could be any ring. The real numbers serve as the most convenient example, however.)
Can we "see", what the mapping does? Can we "see" what $f^5$ does? Given a basis of the two eigenvectors, $(1,-1), (1,1)$ we ...
12
I have always found the standard motivations for eigenvalues to be a little artificial. The primary application for eigenvalues is ultimately diagonalization and there are several ways you could try to motivate diagonalization:
Taking large powers of matrices seems to be a popular one. But its not immediately obvious what this is used for.
An extension of ...
12
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 could (in calculus, DEQns, linear algebra etc.). Now, I think my efforts do have an audience, but not the main audience that I find in my classes. Personally, I've ...
11
When I teach the $L U$-decomposition, mentioned in quid's answer, I use the following mnemonic:
Single $\vec{b}$? Use GE.
Several to do? Use LU.
The point being that if you need to solve $A \vec{x} = \vec{b}$ for several $\vec{b}$ then the $L U$-method is more efficient than setting up a full Gaussian Elimination for each individual $\vec{b}$. In ...
11
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 the full proof, if only the first line or two to give you a hint.
You may also find it worthwhile to get hold of a book on proof techniques, such as Velleman's ...
10
For a real showcase, I recommend a scenario where resonance frequencies play a role.
Suspension bridges are real-world objects which are delicate enough that soldiers are usually not allowed to march over bridges (the German traffic law StVO states this in § 27 Abs. 6). The reason for this is that small, regular excitations with a certain frequency will ...
10
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 the present tools. When a teacher gives an application, the "applied problem" is actually solved in front of the students.
A motivation is a problem or a task ...
10
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"...
So I start from the idea that you can devote at least one hour to this. And it better be a well prepared one, otherwise time will not be enough.
Codes: I do ...
10
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 $A^{-1} = \frac{1}{\textrm{Det}(A)} \textrm{Adj}(A)$, where $\textrm{Adj}(A)$ is something we just now discovered. Hey, the formula for $\textrm{Adj}(A)$ makes sense whether $A$ is invertible ...
9
My personal opinion. Anytime is a good time to teach linear algebra. I'd like to see at least two courses in the curriculum. Ok, more accurately:
elementary matrix theory: solving systems, matrix math, determinants, routine computations of all sorts. Very limited proof emphasis. Here we are merely rounding out a computational core to give better tools to ...
9
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 test?" is a mangled sense of confusion over whether what's currently happening is a skill they're expected to master, or motivation for a skill, or ...
9
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 graphics. Explore which combinations of these transformations always commute, and which sometimes do not commute. In $\mathbb{R}^2$ and in $\mathbb{R}^3$. E.g., ...
9
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 Haskell) where loops and variables are avoided, computation is largely done by recursion, and, most importantly, functions are treated like data and can be passed as ...
8
One of the standard examples is from stability analysis of dynamical systems. In the linear approximation you ask whether the $0$ solution is stable for
$$ X_{j} = A X_{j-1} $$
(discrete time) or
$$ \frac{d}{dt} X = B X $$
(continuous time).
8
One of the main reasons why the cross product is seldomly (to my experience) taught in linear algebra is not only the fact that this particular vector product only works well in $\mathbb{R}^3$ (a specific vector space which isn't the main focus of linear algebra), but that the arsenal of linear algebra provides already a tool to find vectors which are ...
8
So, there is some point where they just propose multiplying the given solution by $t$ and then work out the generalized e-vector of order two condition. Pretty in its way, but I prefer the perspective of the matrix exponential.
Define $e^{tA}=I+tA+(t^2/2)A^2+ \cdots$. It is easy to show that the matrix exponential is a solution matrix and it is an ...
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