Question

Problem 3. Let D be the vector space of all differentiable function R wth the usual pointwise addition and scalar multiplication of functions. In other words, for f, g E D and λ E R the function R defined by: (f +Ag) ()-f(r) +Ag(x) Let R be four functions defined by: s(x)-: sin 11 c(r) : cosz, co(z)--cos(z + θ), and so(r) sin(z + θ), and Wspanls, c Which of the following statements are true: (a) For each fixed θ E R, co, ss& W (b) For each fixed θ R, the set {ca, se} is a linearly independent subset of W 38 MAT2611/101/3/2019 (c) For each fixed 0 E R, the set [co, se is a linearly dependent subset of W, since s where 1 is the constant function 1 1, (d) The set {Sg : θ E R} įs a linearly independent subset of W. only 2. b only 4. c only 5. d only Problem 4. Let GL2(R) be the vector space of 2 x 2 square matrices with usual matrix addition and scalar multiplication, and b cj :a, b.cE R State the incorrect statement from the following five 1. W is a subspace of GL2(R) with basis 0 1 (0 0 0 1-1 00 1 2. W Ker f, where GL2(R)R is the linear transformation defined by: 0 b+ c 3. Given the basis B in option 1.\ coordB( 23)(,2,2) 4, GL2(R) W + V, where: 0 .cER 5. Given the basis B in option 1.\ coordB(

Answer 1

3)-> span [s,c Similarly So ćxekl Ca) is rue b) Fix θ EIR -asın similarly we show aこ fals e Cc) s An swer: 3. Ca)ヂ(W only

w-İh:]: a.b.ceRY a lo obviousy Weary independet a lo Let 0 o B forms basis for LJ 2. Tnu e + 2 coords ( U23] )こ(1, 2, 2) Thu e suppose GL2(R) 0 o c, 2 o W . GL2.COR )=WtV 5) an incorrect statement