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Here too, we consider a vector space *V* and a set *I* (this will be a set of indices, generally of the form
,
but not necessarily).

**Definition 5.3.1**
A family

of vectors in

*V* is a

**generating set** for

*V* if every vector in

*V* is a linear combination of the

's, i.e.

**Example 5.3.2**
We prove that the vectors

and

form a generating set for

.

Let
.
Then:

Thus, any vector

in

is a linear combination of

and

.

**Proposition 5.3.3**
Let

*F* be a generating set for the vector space

*V*. If

*G* is a family of vectors such that

,
then

*G* is a generating set for

*V*.

__Hint:__ Add zeros as coefficients for the extra vectors.

*Noah Dana-Picard*

*2001-02-26*