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
end
function stability(A)

v =   [eigmin(A),eigmax(A)]
if v[1] < -1 ||  v[2] > 1
return 0
else 1
end
end
I am constructing an operator matrix, given by


$u^{m+1} = Au^{m}$

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


$u^{m+1} = Au^{m} = \sum_{n=1}^{N-1} c_{n}^{m} \mu_{n}\zeta_{n}$

$\mu_{n}$are eigenvalues and $\zeta_{n}$are eigenvectors

mu = eigvals(A)
zeta=eig(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);

end

return l
end
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.

So..one 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.

Footnotes

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

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