math 114 (chapter 4-)

every linear combination can be expressed as

a matrix vector product

def span, spanning set

two distinct sets:

• S is a set containing k vectors in Rn

• SpanS is a set of all linear combinations of k vectors in S

how do you show that vector x element of Rn belongs to Span S?

express vector x as a linear combination of vector v1, …, vector vk
if this is not possible, then x is not an element of Span S

thm span and matrix-vector product

→ to check if v is an element of Span(v1,.., vk), we only need to verify that the system Ax=v is consistent

thm span and rank

corollary 4.1.13

for vector v1, v2 element of Rn, U=Span{vector v1, vector v2} is

thm reduction thm

reduction thm proof