A Geometric Review of Linear Algebra [pdf]

At least the way our curriculum was arranged, in first year had a 'simple' linear algebra course that consisted of solving systems of linear equations (like Gaussian Elimination), and 'simple' things with matrices and vectors, which then ended in this rush to teach eigen-decomposition, which we pretty much just learned by rote. I don't believe anyone really had a feel for what we were actually doing.

But the flip side was that we learned eigen-decomposition in time for all the 'fun' applications of it, and by the time our far more rigorous linear algebra course occurred in 3rd year (which ended with SVD), when we went over eigen-decomp again, it was like a brilliant 'OH, I GET ITT' moment for everyone in the class.

They're not missing just judiciously omitted. Actually seeing SVD presented in an introduction is what inspired me to post this.

The SVD approach works for an matrix A in R^{m x n}

   A = UΣV^T

   A = UΛU^T

    {arithmetic}
    |
    {vecops} = {Vector operations}
    {dotproduct}

    {arithmetic}
    |
    {RREF}   = {Reduced row echelon form}

    {arithmetic}
    |  {patience}
    |  |  {dotproduct}
    |  |  |
    {matrices}
    {Mops}   = {Matrix operations}
    {Meqns}  = {Matrix equations}
    {Mmult}  = {Matrix multiplication}
    {Mdet}   = {Determinants}
    {Minv}   = {Matrix inverse}

    {y=mx+b}
    | 
    {linear} = {Linearity}
    {plane}  = {Lines and planes}

    {dotproduct}
    |
    {proj}   = {Projections}
    {basis}  = {Coordinate projections}

    {vectors}
    |
    {VS1}    = {Vector spaces}
    {VS2}    = {Abstract vector spaces}           // knowledge buzz moment
    |
    |  {dotproduct}
    |  |
    {IPS}    = {Abstract inner product spaces}

    {y=mx}
    | {Mmult}
    | |  {function  f:R->R}
    | |  |
    {linT}   = {Linear transformations}
    {VSRREF} = {Vector space techniques for matrices}
    {TasM}   = {Finding matrix representations}
    {basis2} = {Change of basis for matrices}

    {Mmult}
    |  {linT}
    |  |
    {eigV}   = {Eigenvalues and eigenvectors}
       {Mdecomp}= {Matrix decompositions}
       |
    {specM}  = {Special types of matrices}

    {VS1}
    |{plane}
    ||{eigV}
    |||{linT}
    |||||{Meqns}
    ||||||{Mmult}
    ||||||||{Mdet}
    |||||||||
    {invMthm} = {Invertible matrix theorem}          

    {proj}
     |
    {GSorth} = {Gram--Schmidt orthogonalization}

    {RREF}
    |
    {LP}     = {Linear programming}

    {Mmult}
    |{Meqn}
    ||{RREF}
    |||
    {solve}  = {Solving systems of equations}
    {LSaprx} = {Least squares approximate solution}
    {ML}     = {Machine learning}

    {LAoverZ_q}= {Linear algebra over a finite field}
    |||
    ||{ECCs}   = {Error correcting codes}
    |{crypto} = {Cryptography}
    {netcode}= {Network coding}

    {trig}
    |{taylor}
    ||{AVS}
    |||{IPS}
    |||||
    {fourier}= {Fourier series}

   {fourier}
    | {prob}   = {Probability density}
    | |  {LAoverC}= {Linear algebra with complex numbers}
    | |  |
    {quantum}= {Quantum mechanics}

However, I do feel it is necessary: even a limited person can improve by taking all of the juice of the most simple basic maths. I cheat, I use real world out of computer science simple solution in my code.

I noticed the "good student/developers" who were talented tend to solve problem the complicated way. And they developed both an agility and comfort zone out of it. So that my way of helping them saying: «hey man, programming is not about writing code, it is about solving problem with well known, basic understanding.»

You could call it «revenge». I call it irony. And if my words could hurt someone, since I am way more moderated than what I had to cope with, then I consider I am nice.

I won't apologize.

Like knowing non linear problem (like estimating the cost of resource "per user" on the cloud with all their "discrete" rough edged pricing plan) cannot be solved with linear algebra.

The cloud, the cost of IP transit (95th percentile) are defeating by nature any possible estimations. And still I see PhD and geniuses trying to code with linear equations the pricing based on the evolution of users. This is mathematically impossible.

So am I a genius, a dunce or what? I would be glad to know I am wrong, I would stop wondering.

Knowledge don't magically jump into the brain of people.

The most important part about knowledge is not its raw packed informative formalism but the teaching, or letting enough leisure time for people to study by themselves. It is the art of letting people build their own intuitive understanding. No computer, no books, no things can replace a good teacher, or a a good book that suits you.

That is why we need to disagree ;) dialog, and then agree again. Because, maybe dialog is a way of learning.