How do we use null space and column space in real life?

I wrote this blog post a little bit ago.

Suppose that I have some equation that describe heat. It is a partial differential equation. Ok. In this post here.

function operator(s,n)
    v = ones(n-1);
    v = (1-2s)*v
    A = diagm(v)
    v1 = ones(n-2)*s
    v1 = diagm(v1,-1)
    A = A+v1
    v2 = ones(n-2)*s
    v2 = diagm(v2,1)
    A= A+v2
    return A
function stability(A)
  v =   [eigmin(A),eigmax(A)]
    if v[1] < -1 ||  v[2] > 1
        return 0 
        else 1
I am constructing an operator matrix, given by 

[math]u^{m+1} = Au^{m} [/math]

here A is tridiagonal
the solution appears as such in the book 

[math]u^{m+1} = Au^{m} = \sum_{n=1}^{N-1} c_{n}^{m} \mu_{n}\zeta_{n}[/math]

[math]\mu_{n} [/math]are eigenvalues and [math]\zeta_{n} [/math]are eigenvectors

mu = eigvals(A)
s = [.49,.50,.51,.52];
function results(input)
    n1 = 10;
    n = length(input);
    l = zeros(n,2);
    for i = 1:n
        A = operator(input[i],n1);
        zeta = eig(A);
        l[i,1] = input[i];
        l[i,2] = stability(A);
   return l 
l = results(s)
4×2 Array{Float64,2}:
 0.49  1.0
 0.5   1.0
 0.51  1.0
 0.52  0.0

this follows from Gershgorins circle theorem

I outline that we can discretize the solutions to the heat equation and determine if they are stable in terms of a matrix. Then the eigenvalues of the matrix tell us about the stability. Now to get the eigenvalues. We have to use a procedure called Gram schmidt. How? We go to an intermediary step called the Hessenberg matrix. I am not going to post it.

But in Gram Schmidt there is something useful. There is the idea of projectors.

Now when you actually use Gram-Schmidt you are using this idea here.

That we can split spaces into orthogonal components. In doing so we can get nice little orthogonal vectors. simple idea about null space and range allows us to tell you about the solutions to how partial differential equations behave. Why. They use this finite element method So when engineers build buildings they are using this same idea.


Stability of the heat operators by Ryan Howe on Partial Differential Equations

Gershgorin circle theorem - Wikipedia

QR Hessian by Ryan Howe on Linear Algebra

Numerical Stability and Orthogonalization by Ryan Howe on Linear Algebra

Projectors by Ryan Howe on Linear Algebra

The Orthogonal Complement of The Kernal by Ryan Howe on Linear Algebra

Finite element method - Wikipedia

Have you ever given away something you now wish you had kept?

Thanks Rick for a2a. Personally I did not throw away things. My wife, due to my slackness, disposed off a voluminous dictionary from English to Urdu that I used to refer to quite often. I had not placed it close

Could the Faceless Men be hired to defeat the White Walkers?

The faceless men are assassins, not one man armies. They kill their target and get out. Plus they usually kill in such a way as to make it look like a natural or accidental death. On top of that, how would a Faceless

Why are people so anti-tradition?

Because we have progressed so far from our first civilisation and enjoyed too much of modern world comforts, have we not?Traditions vary, from time to time, and it constantly changes. Those who don't adapt will get obsolete and Darwinian mechanism will work its cruel magic