What are some interesting linear functions?

Derivative, integral and by extension of the integral, transforms like Fourier, LaPlace.

I immediately want to say "linearity of expectation" as the cardinal example of a linear thing, but of course talking about L1 on an abstract measure space would go way above their heads.

Perhaps something could be done with a collection of dice or something like that, as a more concrete space of random variables?

The usual Lie theory maps. For concreteness, examples for 2x2 matrices are pretty tractable:

1. conjugation by invertible X on 2x2 matrices, usual basis,

2. same but use usual bases for symmetric + anti-symmetric matrices,

3. Fix any Y. Then X -> XY-YX in bases as above,

4. If physics is on the table, sciflare's example (and its infinitesimal version) leads to angular momentum. You could even hide it in cross-product, curl, etc.

5. Conjugation by unitary on space of skew-Hermitian matrices.

Maybe the adjacency matrix of a graph.

Discrete Fourier transform on a small (4 or 8 component) vector

Multiplication by a specific quaternion, treated as a linear operation on R^4

Applied statistics relies heavily on linear algebra and the methods have a strong geometric interpretation. For example, linear regression involves "inverting" non-invertible matrices in a lower-dimensional space and rotations of the covariance matrix for bivariate normally distributed populations correspond to finding the canonical equation of an ellipse. Both can be found in a lot of books, e.g. "Practical Regression and Anova using R" (free online) or "Applied Multivatiate Statistics" by Richard Johnson and Dean Wichern.

Also, discrete wavelet transforms can be represented as matrix multiplication. Gilbert Strang has a paper here which shows concrete matrices.

The transpose.

Or more generally let them show that any linear map that takes an (m, n) matrix as input and outputs a (m', n') matrix can be expressed as X -> 𝛴i AiXBi

A particularly useful linear projection is the Littlewood-Paley projection P that projects L^2 to functions which are band-limited to frequencies in the interval [2^j , 2^{j+1} ]. This operator has the useful feature that it's easy to motivate (limiting signals to a specific band of frequencies is, very crudely speaking, how a radio works) but its abstract definition is going to be way beyond a second-year course in linear algebra.

But that's no problem. Look at the vector space V spanned by exp(ikt) where k ranges over integers 1, ..., K. Then P has a very obvious form, namely a diagonal matrix with just 0's and 1's. This is, of course, too easy to diagonalize, but there are probably other bases you could put on V which feel natural but in which P does not have a trivial form. An example that I thought of, which seems way too complicated to implement but also a lot of fun, would be to let the basis consist of K different "songs" (= linear combinations of exp(ikt)) and so in some sense the Littlewood-Paley decomposition would be writing a song as a linear combination of other songs.

Write down explicitly the double covering map from SU(2) to SO(3).

I feel like linear maps acting on the vector space of matrices don't get nearly enough in linear algebra courses, so how about the transpose map? Students in every linear algebra course see the transpose of a matrix, but they don't often do linear algebra on the transposition linear transformation (e.g., represent the transpose via its standard matrix, note that transposition is self-adjoint, compute its eigenvalues (-1 and +1), and so on).

Let $p = \sum_{k=0}^{n-1} a_k z^k$ be a polynomial of degree $n-1$ with coefficients $a = (a_0,\ldots,a_{n-1})$. Let $z_0,\ldots,z_{n-1}$ be n distinct points in the ground field (I use the complex numbers). Clearly the map $F$ from $a$ to $(p(z_0),\ldots,p(z_{n-1}))$ is linear and invertible (its inverse can be computed by any desired polynomial interpolation algorithm). I call the linear map $F$ a Fourier transform, parametrized by the choice of points ${z_k}$.

I call those the "coefficient representation" and the "pointwise representation" of a given polynomial. The advantage of the pointwise representation is that it is easy to multiply polynomials pointwise, but to get the coefficients of a product of polynomials, you have to compute a convolution.

From here, you can compute the matrix $M_F$ of $F$, diagonalize it or whatever. An important special case is when ${z_k}$ enumerate the $n$th roots of unity, there it becomes a bona fide discrete Fourier transform, and the computation of the matrix-vector product $M_F x$ can be efficiently accomplished in $O(n\log n)$ arithmetic operations, instead of $O(n^2)$ by traditional matrix-vector product.

Furthermore, the matrix $M_F$ is unitary and hence it diagonalizes, and its eigenvalues have unit modulus. I realize now though that I've never tried to diagonalize it and I wonder what the eigenvectors are.

Some more finite-dimensional linear applied things:

• The discrete one-period model for option pricing (M. Dothan - Prices In Financial Markets).
• The simplex method from linear optimization (sorry, I haven't looked for good English resources on the subject).
• Some graph optimization algorithms can be computed via matrices over dioids (M. Gondran, M. Minoux - Graphs and Algorithms).

Maybe something from Linear Sketching like Johnson-Lindenstrauss.

Maybe by using the linear differential equation(of order n) i.e

a_ny^n + ........ a_1y + a_0 = 0

and using that as a linear operator(in H(C^n(R), where that is the space of real valued function which can be differentiated at least n-times)

So T(y) = a_ny^n + ........ a_1y + a_0

and Ker(T) is the space of solutions

The evaluation map for different domains:

1. from the space of polynomials K[X] over a field K to the field K
2. from K[X] to the vector space of square matrices over K
3. from the Hom(V,W) to W for vector spaces V and W

This blog post about linear from R to R is pretty interesting: http://www.cofault.com/2010/01/hunt-for-addictive-monster.html?m=1

On second thought I suppose these aren't actually R-linear, they just satisfy the functional equation f(x+y)=f(x) + f(y). Still interesting though

There's some basic differential equations stuff with matrix exponentials:

If u: R^n -> R^n and A is some matrix, then the equation

du / dt = Au

with initial data u = u_0 has as solution

u(t) = exp(tA)u_0

Think of this as a system of ODEs. If you're feeling bold you can throw around words like "semigroup" or "one-parameter group of transformations".

As a specific example, let

dv / dt = w dw / dt = -v

Then A = [[0, 1], [-1, 0]] and we can set it up above as

u = exp(tA) u_0

but also notice that the system is really v'' = -v, so v = e^(it)v_0 also works. This opens up a potentially interesting connection.

Kind of boring but the q derivative

D_q f(x) = f(qx)-f(x)/(q-1)x

and the forward difference

D f(x) = f(x+1) - f(x).

If you're talking about Jordan form you might be passed it, but you could also talk about the canonical basis for D being the falling factorial.

Another one might be summation.

Google’s PageRank algorithm is ultimately based on calculating the eigenvectors of a particular matrix that encodes all references from every webpage to every other webpage. More generally, you can calculate what a Markov chain would look like in steady-state conditions (if it exists) by calculating eigenvectors - and you can model quite a lot of real life things with that.

Another thing that was really cool for me when I was first learning linear algebra was writing the Fibonacci series as an iterated matrix multiplication, from which you can then get a closed form for the nth term of the series by using eigendecomposition (and having the golden ratio and its inverse show up in the process).

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