Please answer the following question

Let two bases for a two dimensional space be S : {u1,u2} T : {V1,v2}. Say we have the coordinate of a vector w W.r.t 8, hence we have _ a1 [Wis [ a2 ] W : alul + (12112. such that What is the coordinate vector of w w.r.t. T? Let two bases for a two dimensional space he 3 {'11.~'12} T {V1,V2}. Say we have the coordinate of a vector w w.r.t S._ hence we have such that W = (111.1] + qug. What is the coordinate vector of w w.r.t. T3 First write u11u2 w.r.t. T1 say U] _ Clvl +114\"? \"'2 J1V1+J2V2 Then w = Gall1.] +112\"? = 1iC1V1+f>2V2l+H2id1V1+ \"level = [:1ch +'12d1lvl +i'1202 +HZJ2JV2 Hence 15192 + (1.ng C] d] [.1] (3'1 112 [1.2 I Which is very nice. since this means that1 whatever [w]5,- is1 we need only multiplyr it on the left with _ c] d1 P _ i e d2 i to get [w]g-. So the matrix efciently changes one coordinate vector to another. C1 ] is [u1]r;~ [since 111 = c1v1+c2V2j and [ d1 ] {'2 (E2 is [112];- [since I12 = d1V1 + dying). Hence._ in order to nd P we need only write each of the has-is elements of S in terms of the basis T. The coordinate vector of the rst basis element found is then column 1 of P1 that of the second column 2 of P and so on. [W]?' = [ o1c1+agd1 ] which can be written as: We now note that the column [ Same reasone, Bari's = B= 3bibk3. linear combination 5 - 5, bit . .foxbr coordinate vector CS] = 10, ...ck] here we want CST p smread of wwrt Te [ W ] wis of the form a, di tazy, we want to write ordered list of no: in terms of pernicular ordered basis . coordinate vector of w. was to T w = a , lit a 2 4 2 . sur+ 1 23-5 is bases of & diamensonal, we want cordinslivertor wir toT. T= 30, V 23 . 30 If write ving W.r to T, say , up = CIV, + ( 2 V2 ( linear comb" ot basis ) U 2 = div, tdave put this in w = alllit a, 42 3 a ( c , v , t cong ) + ap ( d, v , + d 2 v 2 ) combine v, terms 9 2 terms . fa s , + a z d , ) v , + ( a , ( 2 + 9 , de ) v 2 [ this is Of the form by v , + by v z ? here, by = arc , + and 1 b 2 = a 1 ( 2 + 9 2 d 2 linear combination of vector in terms of a particular ordered basis . Laz c2 + and 2 = ( c , d ) ] , / an - c, at diaz ) ( 2 d z Jaz = Czart dzaz PAGE