>I'd go a bit further and say that if you're not comfortable with the basics of mathematical proofs, then you're not ready for the subject of linear algebra
I can't write a proof to save my life, but I'm going to keep using linear algebra to solve problems and make money, nearly every day. Sorry!
We had this discussion about Data Science years ago: "you aren't a real Data Scientist unless you fully understand subjects X, Y, Z!"
Now companies are filled to the brim with Data Scientists who can't solve a business problem to save their life, and the companies are regretting the hires. Nobody cares what proofs they can write.
There are (at least) two different things we're calling "linear algebra here", roughly speaking one is building the tools and one is using the tools.
The mathematicians need to understand the basics of of mathematical proofs to learn how to prove new interesting (and sometimes useful) stuff in linear algebra. You have to do the math stuff in order to come up with some new matrix decomposition or whatever.
The engineers/data scientists/whatever people just need to understand how to use them.
You don't need to know how to build a car to drive one. The mathematicians are building the cars, you're using them.
I can't write a proof to save my life, but I'm going to keep using linear algebra to solve problems and make money, nearly every day. Sorry!
We had this discussion about Data Science years ago: "you aren't a real Data Scientist unless you fully understand subjects X, Y, Z!"
Now companies are filled to the brim with Data Scientists who can't solve a business problem to save their life, and the companies are regretting the hires. Nobody cares what proofs they can write.