Linear Algebra, 5th Edition

1.6 Bases and Dimension

We saw in Section 1.5 that if S is a generating set for a subspace W and no proper subset of S is a generating set for W, then S must be linearly independent. A linearly independent generating set for W possesses a very useful property—every vector in W can be expressed in one and only one way as a linear combination of the vectors in the set. (This property is proved below in Theorem 1.8.) It is this property that makes linearly independent generating sets the building blocks of vector spaces.

Definition.

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

Example 1

Recalling that $\text{span }(\varnothing )=\left\{0\right\}$ and $\varnothing$ is linearly ...