The wall has a name: change of basis

This is a continuation of the linear algebra thread I started here. Linear algebra has actually been a pleasant ride so far. If I were only in need of knowing how to crank out the mechanics, I'd be pretty set.

I feel though, that getting a core understanding of what matrix operations mean to quantum computing and quantum theory in general is going to be critical going forward. Thus, I have to dig deeper. 3Blue1Brown has been essential here. In the earliest sections, I've been able to make my way through on simple mental perception of transformation in space, especially in regard to vectors and simple matrix multiplication.

Finishing Quantum by Manjit Kumar was especially helpful, as it revealed that many founding quantum theorists were unfamiliar with the idea that AxB doesn't necessarily equal BxA. Linear algebra, while fully formed at the time, existed as a theoretical solution in search of a quantum problem.

Pulling out that matrix A's row multiplied against matrix B's column meant that order mattered. It's wild to think that some theorist was able to look at a real world problem and pull matrix multiplication out of the air as a solution.

Change of basis

So what's the wall? It's the basis. 4x4 matrix multiplication is simple, though monotonous. I can crank out eigenvalues without much of an issue. Many simple matrix transformations are simple enough to mentally visualize, as if they're stretching silly putty.

But change of basis, this one is going to take a while. As I said in the beginning, if I were only taking this as a university class, I could "shut up and calculate" and get a solid B. But this is foundational, and thus important to get a solid understanding of matrix transformations.

I'm generally following Strang's Introduction to Linear Algebra with a decent amount of outside material. Current status:

• Vectors and Matrices: down cold
• Linear Equations: down cold
• Four Subspaces: conceptually OK
• Orthogonality: mechanics are OK, not so much on theory
• Determinants: mechanics are OK, not so much on theory
• Eigenvalues and Eigenvectors: mechanics are OK, not so much on theory
• Singular Value Decomposition: to be done
• Linear Transformations: to be done, probably the next huge wall
• Linear Algebra in Optimization: to be done
• Learning from Data: to be done, theoretically not a huge lift given background

You can see my quantum competency project on my git here. I also estimated my hours spent per item. By the time I'm done, I'll be able to identify approximately how many hours it takes a data professional to upskill to applied quantum.

I believe the industry assumption is about three to six months for fundamentals (ie. where I’m at now), and 1.5 total years for job readiness through a STEM background. I'm expecting that I’ll be mostly on track against these assumptions.