I would also add that one fundamental aspect of linear algebra (that no one ever taught me in a class) is that non-linear problems are almost never analytically solvable (e.g. e^x= y is easily solved through logarithms, but even solving xe^x=y requires Lambert’s W function iirc). Almost all interesting real world problems are non-linear to some extent, therefore, linear algebra is really the only tool we have to make progress on many difficult problems (e.g. through linear approximation and then applying techniques of linear algebra to solve the linear problem).