Learning about linear combinations and spans:
A linear combination of a list of vectors is a vector of the form: ${latex.inlinea_{1}v_{1} + ... + a_{m}v_{m}} where ${latex.inlinea_{i} \in F}
How does this relate to sums of vector spaces? Each vector in the sum of vector spaces is itself a linear combination of those vectors.
The Span is the set of all possible linear combinations of vectors in a list.
How do we relate this to the sum of Subspaces? Well, suppose we have two subspaces V1 and V2. Then each one of those has a basis(we get there later). Call the basis of V1: {v11, v12, v13} and the basis of V2: {v21, v22, v23}. Each of the vectors in v1 are linear combinations of v11, v12, v13. Each of the vectors of V2 are linear combinations of v21, v22, v23. Each of the vectors of V1 + V2 are written as linear combinations of v11, v12, v13, v21, v22, v23.
A finite dimensional vector space is a vector space which is spanned by a finite list of vectors. ie there is a finite list of vectors that can be linearly combined to express each v in V.
A list of vectors is called linearly independent if the only choice of ${latex.inlinea_{i}} that makes ${latex.inline0 = a_{1}v_{1} + a_{2}v_{2} + ... + a_{m}v_{m}} true is all a’s being 0.
Some Theorems and Counterexampels
2.6 The span of a list of vectors in V is the smallest subspace of V containing all vectors in the list.
2.19 (The key finding of this chapter) aka the linear independence lemma: If there is a linearly dependent list of vectors, they must span something. Imagine taking all of the combinations. Well, inside that list of vectors there is at least one vector that can be written as a linear combination of the others, so it is in the span of the others. That one can be removed, and the span remains unchanged. In other words, intuitively, a linearly dependent list of vectors contains some extra unnecessary information about the v that we can remove.
NOTE: Can we prove ourselves that the span of a list of vectors is always a vector space?
2.22 The length of a linearly independent list of vectors is less than or equal to the length of a spanning list.
2.25 Every subspace of a finite dimensional vector space is finite dimensional.