Linear Independence and Spanning Sets

Question 1

What is a fact about LD and LI and spanning sets?

Answer 1

A set {u1,···,um} is LD if and only if there is at least one
vector uk which is in the span of the rest. Equivalently,
{u1,···,um} is LI if and only if none of the vectors is a linear combination of the others.
Note: This does not mean that every vector is a linear combination of the
others in a LD set.

Question 2

What is a fact about reducing LD sets?

Answer 2

If W = Span{u1,···,um} is a subspace of a vector space V and {u1,···,um} is LD, for instance u1∈Span{u2,···,um}, then W = Span{u2, · · · , um}.

Question 3

What is a fact about enlarging LI sets?

Answer 3

Suppose {u1, · · · , um} is a LI subset of a vector space W. For any v ∈ W we have {v, u1, · · · , um} is LI ⇔ v∉Span{u1, · · · , um}.

Question 4

Summarize the fact about reducing LD sets?

Answer 4

Any LD spanning set can be reduced without changing the subspace they span by removing a vector which is in the span of the rest.

Question 5

Summarize the fact about enlarging LI sets?

Answer 5

Any LI set in a subspace W, which does not span W, can be made into a larger LI set in W by throwing in a vector which is not in the span of the rest.

Question 6

What is a fact about LI sets in R^2?

Answer 6

Any LI set in R^2 has at most 2 vectors and any spanning set of R^2 has at least 2 vectors. Any LI spanning set of R^2 has exactly two vectors.