- 1.3
- 1.4

Definition: A subset

*W*of a vector space*V*over a field*F*is called a subspace of*V*if*W*is a vector space over*F*with the operations of addition and scalar multiplication defined on*V*.Problem: Normally, there are 8 properties you need to check. But it turns out you only need to check 4 of them. Which 4? Why?

- Problem: (Theorem 1.3) Let
*V*be a vector space and*W*and subset of*V*. Then*W*is a subspace of*V*if and only if the following three conditions hold for the operations defined in*V*.- 0 ∈
*W*. *x*+*y*∈*W*whenever*x*∈*W*and*y*∈*W*.*c**x*∈*W*whenever*c*∈*F*and*x*∈*W*.

- 0 ∈
- Problem: Same problem as before, but replace conditions 2 and 3 with
*c**x*+*y*∈*W*whenever,*x*,*y*∈*W*and*c*∈*F*.

Problem: Give an example of a vector space

*V*and a subset*W*of*V*such that,*W*is a vector space but*W*is not a subspace of*V*.Problem: Show that the intersection of 2 subspaces is a subspace.

Definition: Let

*V*be a vector space and*S*a nonempty subset of*V*. A vector*v*∈*V*is called a linear combination of vectors of*S*if there exists a finite number of vectors*u*_{1}, …,*u*_{n}in*S*and scalars*a*_{1}, …,*a*_{n}in*F*such that*v*=*a*_{1}*u*_{1}+ ⋯ +*a*_{n}*u*_{n}.Problem: We denote the set of all linear combinations of

*S*by*s**p**a**n**S*. By convention, we define the span of the empty set to be the trivial subspace {0}. Prove that*s**p**a**n*(*S*) is always a subspace.Problem: Let

*S*⊆*T*be sets inside of a vector space*V*. Prove that*s**p**a**n*(*S*) is a subspace of*s**p**a**n*(*T*).Problem: Prove that

*s**p**a**n*(*S*) is a the smallest subspace containing*S*. (This gives an alternative definition of*s**p**a**n*(*S*) that turns out to be quite useful!)