Question

(13) B = = {{]] • ]} and c = {{=} (1) are both bases for R2. (a) (5 points) Find the matrices P. and P CAB вес (b) (5 points) What is [7} if (c = [-] ? And what is v in standard coordinates? (.) (5 points) What is [v]e if [W]s = [2] ? And what is V in standard coordinates?

Answer 1

as a We express [] and [] Vectors of e linear combination of Now , [-]] •[11-10 on. -1 - 2 1 on, X JO o -1 Solving we get; y = -1 -x+y = 2 → -*= 3.>> X=-3 Now , [:] - [14]+ y [1] 1 on OD 2 1 Solving we get, y=-1 -x+y = 1 + 2 - 1 = 1 x = -2 So, [:] - 4-2 [] + (-1) [1] [1] (-2 [1] +(-1) 3

So, P. CEB linear as a Now, X 2 • Neos, coe express [-2] and [1] Combination of vectors of B. - [30-03) - [A] Solving we get, y=1. +y 2x +y = -1 - 2x + 1 = -1 => x = -1 32 Nese 1] +9[:] - - D:40:1-BJ -BAR Solving weget, y=-3 and 2x+y = 1 on, 2X = 4 m, *= 2 (-1) - (-2) [] +(1) [:] = (2) [:] + (-3) So, 2 So, P [ Вес 1 -3 (2)

Giver, [U].-33) Novo, [0] = 0 (53] + [0] Now, Foom (2) we get, B [1-01-13) (Ars) (C) Given, [0]. - [8] So. 7 = (29)[]] + [3] - [A From (1) we get, - -2 [U]. - [3 :]]-[] (An)