I always thought it was simpler to explain by using the only trick linear algebra has: switch to a basis where your problem is easiest, then switch back.
Sylvester's law of inertia proves existence + gram-schmidt constructs that change of basis
Sylvester's law of inertia proves existence + gram-schmidt constructs that change of basis
Do you have any additional linear algebra tricks?
But a lot of it is simply exploiting linearity to reduce to a working in the most convenient subspaces. Finding and constructing them is a major task.
The matrix is simply the model of the linear space. Coordinate free-ness lets you get away from coefficient chasing and tedium to focus on the real stuff.
Esp important in infinite dimensional vector spaces where many theorems cannot rely on a basis existing as it is equivalent to the axiom of choice.
Solving indefinite integrals by substitution for example. Justified by the chain rule.