5 Vector Spaces

Everything so far has concerned columns of numbers. This section keeps the operations and throws away the numbers.

A vector space is any set on which addition and scalar multiplication behave as they do in \(\mathbb {R}^n\). The gain is reach: polynomials, matrices, continuous functions and sequences all satisfy the same axioms, so every theorem proved here applies to all of them at once, and a fact established for \(\mathbb {R}^3\) need not be re-proved for the space of polynomials of degree at most two.

The work of the section is to make “size” precise. Spanning says a set is large enough to reach everything; linear independence says it contains no redundancy; a basis is a set that is both, and the exchange lemma (Lemma 5.4.3) shows every basis of a space has the same number of elements. That number is the dimension, and it is the invariant the rest of the course counts with.

To define a vector space we need first the notion of a field.

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