If b l c, :: :: I = - 2. find ('2 c ) {I I - ~(/ ) a, a, iJl - ~b) b, b, , C l - 1:c) c, ' )

1 MATH 205 LINEAR ALGEBRA SUPPLEMENT STEVEN H. WEINTRAUB 1. Coordinates and Change of Basis Deﬁnition 1.1. Let B = {v ,v1,..2,v } bn a basis for the vector space V . Let v ∈ V so that v = c 1 1 c 2 +2▯▯▯ + c vn n for unique scalars 1 ,2 ,...,n . The scalars 1 2c ,...nc are called the coordinates of v in the basis B, and the vector c1 c2 [v]B= . . c n is called the coordinate vector of v in the basis B. n Deﬁnition 1.2. The standard basis for R is E =n{e ,e1,.2.,e } wnere e is ihe vector in R with i th entry equal to 1 and all other entries equal to 0. (We will generally abbreviate E no E when there is no possibility of confusion.) 1 0 0 For example, the standard basis for R is E = { 0 , 1 ,0 }. 0 0 1 Lemma 1.3. Let B = {v ,v 1...2v } benany basis of V . (a) [0B = 0. (b) [v ] = e . i B i Proof. Part (a) is just the observation that 0 = 01 + 0v 2 ▯▯▯ + 0v n Part (b) is just the obser