Vector spaces and linear independence

Linear algebra starts with arrows in space, then throws the arrows away. What actually matters is that you can add two objects and multiply one by a number, and that these operations obey the familiar rules. Any set with that structure is a vector space, and the theory applies to it unchanged.

That abstraction is not decoration — it is the whole payoff. Polynomials of degree at most 3 form a vector space: add two and you get another, scale one and you get another. So do continuous functions on an interval, solutions of a linear differential equation, and signals in a Fourier analysis. Every theorem proved about arrows transfers to all of them, which is why "the space of solutions is two-dimensional" is a meaningful and useful statement about a differential equation.

Independence is the organising idea. A set of vectors is linearly dependent if one of them can be built from the others, in which case it carries no information the rest do not already provide. An independent set that also spans the space — every element is some combination of them — is a basis, and every basis of a given space has the same number of elements: the dimension. Two warnings follow. Having the right number of vectors is not enough: three vectors in space that happen to lie in a common plane are dependent and span only that plane. And a basis is never unique — there are infinitely many, and choosing a convenient one is a large part of what makes linear algebra useful, since the right basis can turn a hard problem into an obvious one.