Linear span

• Suppose , where is some subspace of . Then . Then, is a subspace of

Proof: Pick . , for some . , for some . is a scalar multiple of , hence Pick . is a scalar multiple of . . Thus, is a subspace of . First step: Show Pick . . But and is a subspace. Since is closed under scalar multiplication, it means is a subset of .

• is really the definition of a line
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Definition: Linear span. Let be a non-empty subset of . Define the linear span of in as

• This is essentially picking finitely many vectors from and consider any linear combination of the following vectors
  • This space is a subspace of
  • The linear span of two linearly independent subspaces will not be subsets of each other

Proposition: For any subset , the linear span is a subspace of Proof: Pick . Then, where . is a linear combination of elements of . Thus, . Pick then and where So, is a linear combination of the elements of . Thus, . Hence, is a subspace of

Example: Say that we have a set , where . We want to show that . To do that, we have to show that 1) , and 2) . Case 1 is immediate, since we know that . For case 2, pick . We have to work to show that . . . We need to find such that Solving for , we get that . Now, we can describe any with the linear span of . . Thus,