- Linear Combination: Let and . Then is said to be a linear combination of vectors and .
- like we use
β to scale vectors
- For example, we can use vectors to represent a plane by scaling them with
- Span: Let . Then the span of these vectors, Span , is said to be the set of all vectors that are a linear combination of the given set of vectors.
- Let . means we can use the linear combination of vectors u and v to represent all of the vectors .
Definition: A spanning set of a subspace S is a set of vectors such that Span() = S.
- For example:
Definition: Let . Then this set of vectors is said to be linearly independent if implies that . A set of vectors that is not linearly independent is said to be linearly dependent.
- In other words, the only solution for is
- For example: is linearly dependent.
- Because the set can be represent with . We can do: to make the linear combination to be 0. And don’t have to make all .
- In other words, doesn’t give us any new dimension, still the same as .
- is linear independent set.
- Also, we know that two vectors with different directions can span a plane. So if we add any vectors to , it will be linear dependent set.
Theorem: Let the set of vectors be linearly dependent. Then at least one of these vectors can be written as a linear combination of the others.
- In other words, the dependent vector can be written as a linear combination of the other n−1 vectors.
Theorem: Let and let . Then the vectors are linearly independent if and only if .
Definition: A basis for a subspace S of is a set of vectors in S that
- is linearly independent and
- Spans S.
- Basis is the minimum set of vectors that spans the subspace.
- Let . Then are linear independent,
- And all of the linear combinations of can get all of the possible components of . And each member of U can be uniquely defined by a unique combination of .
Theorem: Let S be a subspace of and let and both be basis for S. Then . In other words, the number of vectors in a basis is unique.
Definition: The dimension of a subspace S equals the number of vectors in a basis for that subspace.
- For example:
- then is the basis of .
- the dimension of null space of A = 3, which also equals to the number of non-pivot columns of .
- the dimension of A = 2, which also equals to the number of pivot columns of .
Definition: Let . The rank of A equals the number of vectors in a basis for the column space of A. Denoted by .