noBSLAnotebooks

e? In high school math we learn about proportionality relations, i.e. y = mx, where the output y depends on the input x multiplied by coefficient m (the slope if you think geometrically in the xy-plane). Extending the notion of proportionality to transformations with n inputs and k outputs, instead of the single slope m you need k*n coefficients to describe the proportionality relations between input component j and output component i. Linear models are pretty good in the bang-for-your-buck metric for math models since: (1) k*n params for an R^n --> R^k transformation is reasonable amount of parameters, and (2) using "tomography*" you can easily estimate each of the coefficients. This is why LTs are used in many fields of science/computing use linear transformations (Biology, Chemistry, Economics, Statistics, NNs, etc.). tomography*: input n "probing inputs" to T e1, e2, ..., en and record the outputs T(e1), T(e2), ..., T(en) (each of the outputs is a k-dimensional vector) --- if T is a linear transformations, then the info you've collected is enough to know all the k*m coefficients of the linear transformation (<=> entries of the matrix). 2/ From a theoretical point of view, I think teaching linear transformations and matrix-vector products (the boring row-times-column arithmetic rules) is a really good thing since it introduces learners to representation theory. You have one thing in math land y = T(x) and another thing in math land y = Mx and you know their behaviour and properties are identical (isomorphic?). This means you can understand the properties of one of the things by studying the properties of the other thing, e.g., Ker(T) <=> Nullspace(M). For me, this first contact with representation theory concepts feels like a really valuable thing to have. (A good knowledge buzz moment to get learners more interested in learning math). And it's not just linear transformations and matrices that have a "is a representation of" relationship between them. There are lots of representations in LA: - vectors <-> coordinates - system of equations <-> matrix equation - row ops <-> elementary matrices - linear transformations <-> matrix-vector products - composition of linear transformations <-> matrix-matrix products - graph <-> adjacency matrix - conditional prob p(y|x) <-> matrix whose columns are p(y|x_i) - function in time <-> Fourrier coefficients - quantum state <-> vectors with complex coefficients - quantum operation <-> unitary matrices - quantum measurement <-> list of projection matrices that sum to the identity In particular for devs, it's a really easily transferrable analogy. The linear transformation T is the "spec" while the matrix M and matrix-vector product rules together represent a particular implementation of the spec. See `T` and `T_impl` near the bottom of this notebook https://github.com/minireference/noBSLAnotebooks/blob/master... (binder https://mybinder.org/v2/gh/minireference/noBSLAnotebooks/049... or colab https://colab.research.google.com/github/minireference/noBSL... ) 3/ From a pedagogical point of view, if we can deliver the "representation theory buzz" from linear algebra in high school, then this will be a good chance to review some important high school ideas: Integer representations: - integers in terms of decimal digits with place value a = dn...d3d2d1d0 = dn*10^n + ... d3*1000 + d2*100 + d1*10 + d0 - integers in terms of prime factorization a = 2^a2*3^a3*5^a5*... Rational representations: - fractions as m/n where m in Z and n in N\* - reduced fractions as m/n where m in Z and n in N\* where GCD(m,n)=1 So all in all, if we were to define LA as "representation theory knowledge buzz," and someone asks me if I recommend learning LA before CALC, I'd say yes.*

= [y1,y2,...ym]. If you allow arbitrary functions like quadratics, sine, cosine, exp, etc. there would be way too many different ways to map vec(x) to vec(y). Too many ways... This is where the "linear" part of linear algebra comes in. Instead of arbitrary input output transformations vec(x) --> vec(y), what if we restrict oursleves only to relationships where the output yi is some linear combination of the inputs x1 x2 .. xn, where linear combination means y1 = m11*x1 + m12*x2 + ... + m1n*xn in other words, take the outputs are equal to the inputs times some proportionality coefficients mij. The other output coefficients y2...ym are computed in the same way with different coefficients y2 = m21*x1 + m22*x2 + ... + m2n*xn ... ym = mm1*x1 + mm2*x2 + ... + mmn*xn thus the overall map vec(x) --> vec(y) has a total of m*n coefficients mij. This map between inputs vec(x) and outputs vec(y) is the vector-equivalent of the single-variable function y=mx (proportionality relationship, or line with slope m passing through the origin if your prefer to think of it geometrically). We call these maps "linear transformations" and they have really nice properties. If you study linear algebra you'll learn all about them. Computationally (usefully), geometrically (Whoooa moments), and theoretically (knowledge buzz). And that's just the math part! Once you start to look into applications, it goes really far... Others in this thread have already mentioned numerous areas of applications in CS areas, and it's the same in biology, chemistry, economics, etc. TL;DR: linear algebra is good stuff! PS: Here is a short video where I explain more about why linearity [normally written as f(ax1+bx2) = af(x1)+bf(x2)] is so important: https://github.com/minireference/noBSLAnotebooks#ch2-lineari...