span

(3Blue1Brown) (CliffNotes)

Let be a subset of a vector space . The span of is the set of all linear combinations of vectors

It answers the question “What are all the possible vectors we can reach using only fundamental operations (addition and scalar multiplication)?”

is the SMALLEST subspace of that contains all vectors of

Why SMALLEST?: Because any vector that is a linear combination of the rest (=linearly dependent) could be discarded from the span without affecting it.

Proof: Span is subspace of

Proof: If is a subset of a subspace of a vector space , then is a subset of .

//TODO

Problems

Check if a set of vectors span the vector space

(LTC) The set of vectors span the vector space if every vector  in  can be written as a linear combination of vectors in 

Whether a vector is in the span

A vector is in the span of other vectors if it could be written as a linear combination of those vectors

Example

1. ?

2. ?

So both vectors are in the span of

Find the span of a set of vectors

1. Define the span
   1. as a subspace (to find out its dimension)
   2. as a set of all linear combinations of these vectors
2. Find the normal vector
3. Using properties of a vector space