Linear Algebra

Section V. Change of Basis 241can calculate Ĥ = RepˆB, ˆD (h) either by simply using ˆB and ˆD, or else by firstchanging bases with RepˆB,B (id) then multiplying by H = Rep B,D(h) and thenchanging bases with Rep D, ˆD (id).This equation summarizes.Ĥ = Rep D, ˆD(id) · H · RepˆB,B(id)(∗)(To compare this equation with the sentence before it, remember to read theequation from right to left because we read function composition from right toleft and matrix multiplication represents composition.)2.1 Example The matrix(cos(π/6)T =sin(π/6))− sin(π/6)=cos(π/6)(√ )3/2 −1/21/2 √ 3/2represents, with respect to E 2 , E 2 , the transformation t: R 2 → R 2 that rotatesvectors π/6 radians counterclockwise.( ) ( √ )1 (−3 + 3)/23t π/6−→(1 + 3 √ 3)/2We can translate that to a representation with respect to( ) ( ) ( ) ( )1 0−1 2ˆB = 〈 〉 ˆD = 〈〉1 20 3by using the arrow diagram and formula (∗) above.R 2 wrt E 2id⏐↓R 2 wrt ˆBt−−−−→TR 2 wrt E 2t−−−−→ˆTid⏐↓R 2 wrt ˆDˆT = Rep E2 , ˆD (id) · T · RepˆB,E 2(id)Note that Rep E2 , ˆD (id) is the matrix inverse of Rep ˆD,E 2(id).( ) −1 (√ ) ( )RepˆB, ˆD (t) = −1 2 3/2 −1/20 3 1/2 √ 1 03/2 1 2((5 − √ 3)/6 (3 + 2 √ )3)/3=(1 + √ √3)/6 3/3Although the new matrix is messier, the map that it represents is the same. Forinstance, to replicate the effect of t in the picture, start with ˆB,( ) ( )RepˆB ( 1 1) =3 1ˆB