Linear Combinations and Span Flashcards
What is a linaer combination and when is the combination trivial?
A vector in the form = a1v1+a2v2+…
Trivial when all scalars are 0
What is the span of a set of vectors S?
Let S be susbet of V. The span of S is the set containing all linear combinations of the vectors in S.
When does a list of vectors B in a vector space V, span V?
If every vector v in V is a linear combination of the vectors from B
How do we calculate if a set of vectors spans a space?
Form linear combinations ,set up and solve system.
If there is a solutin (whether infinite or not), then it spans. If no solutuion, then does not span V.
What does it mean when a set of vectors S span a vector space V for a specific basis vector, and how is it calculated?
The vectors in S are able to cover the entire space V in the direction of that particular basis vector.
Calculated by writing basis i as linear combination of vectors in given set (for each vector in basis)
What is R^n spanned by?
e1, e2, …, en
where e is unit vector with unit at n
What is the span of a subset of V?
If S is the subset
The span of S is a subspace of V
How can vector spaces be defines using subspaces?
Defined as subspace of those vectors that can be written as a linear combination of some other vectors
Therefore the span of a set of vectors will be a subspace of the vect
How can we determine whether a list of vectors, C, spans V if one already knows that some other list B spans V?
If B spans V and each vector in B is a linear combination of those from some other list C, then C spans V.
When is a list of vectors in a vector space, linearly independent? (And what special case will it never be independent?)
Linearly independent if the linear combination of the set of vectors = 0, only has the trivial solution (where all scalars are 0).
Otherwise if it has a different solution, then dependent.
If the list contains the zero vector, then it will never be independent.
When is the list of one vector dependent?
When the vector is zero vector
When is a list of 2+ vectors linearly dependent?
One of the vectors in the list is a linear combination of the other vectors in the list.
What is the Steinitz exchange lemma?
If L_m is a linearly independent list of vectors in space V and S_n spans V
- m <= n
How do we prove that a set of vectors is linearly dependent?
Assume there is some linear combination and show that not all scalars involved are 0
