Chapter 1.6 Definitions (Bases and Dimension)

All definitions in Chapter 1.6

Basis

A _____ for a vector space V is a linearly independent subset of V that generates V. If B is a basis for V, we also say that the vectors of B form a basis for V.

Theorem 1.8

Let V be a vector space and u1, u2, …, un be distinct vectors in V. Then B={u1, u2, …, un} is a basis for V if and only if each v∈V can be uniquely expressed as a linear combination of vectors of B; that is, can be expressed in the form

v = a1u1 + a2u2 + … + anun

for unique scalars a1, a2, …, an

Theorem 1.9

If a vector space V is generated by a finite set S, then some subset of S is a basis for V. Hence V has a finite basis.

Theorem 1.10 (Replacement Theorem)

Let V be a vector space that is generated by a set G containing exactly n vectors, and let L be a linearly independent subset of V containing exactly m vectors. Then m≤n and there exists a subset H of G containing exactly n-m vectors such that L∪H generates V.

To be continued….

To be continued….