MATH125 Lecture Notes - Lecture 9: Row Echelon Form, Scalar Multiplication, Linear Map

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: Vladimir Chernousov

Document Summary

In practice, all subspaces occur as a span of vectors, let h=span(v1, v2, vp) If h has a basis v1, v2, vs, then every vector can be written uniquely as a linear combination where w=c1v1+ c2v2+csvs. If we subtract these two equalities, we are left with: 0=(c1-c1")v1+(c2-c2")v2++(cs-cs")vs. And if this equals 0 and they are linearly independent, we only have the trivial solution, meaning that: c1-c1"=0 and c2=c2"=0 and cs-cs"=0, and therefore all scalars have the same value and there is only the unique solution. Definition: let h be a subspace in rn let b be its basis. B={u1, u2, us}, let w be a vector=c1u1+c2u2++ csus then the scalars are called the coordinates of w relative to basis b and is denoted by [w]b=[c1,c2,,cs] B is a basis of h ={v1, v2, ,vp} =c1v1+c2v2++cpvp, and the scalars are called the coordinates of vector v relative to b, denoted [v]b =[c1, c2, ,cp]