Page 195 - 35Linear Algebra

9

Subspaces and Spanning Sets

It is time to study vector spaces more carefully and return to some funda-
mental questions:

1. Subspaces: When is a subset of a vector space itself a vector space?
(This is the notion of a subspace.)

2. Linear Independence: Given a collection of vectors, is there a way to
tell whether they are independent, or if one is a “linear combination”
of the others?

3. Dimension: Is there a consistent definition of how “big” a vector space
is?

4. Basis: How do we label vectors? Can we write any vector as a sum of
some basic set of vectors? How do we change our point of view from
vectors labeled one way to vectors labeled in another way?

Let’s start at the top!

9.1 Subspaces

Definition We say that a subset U of a vector space V is a subspace of V
if U is a vector space under the inherited addition and scalar multiplication
operations of V .

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