第五讲习题解答(讲义) 1.判断下列各变换是否是线性变换 解:()是,对21,2∈R和c∈R,有 x1+x2+k(y+y2) y1+y2 x1+91+ k(cy1 (b).不是,对wx,y∈V,有 o(x+y=afa+a=o(x)+o(y (c).是,对vf1(x),f2(x)∈F[]h和vc∈F,有 a(f1(x)+f2(x)=f1(x+1)+f2(x+1)=a(f1(x)+a(f2(x) o(efi(a))=cfi(a+1)=co(i(a)) (d).是,对vf1(x),f2(x)∈F[x]l和ve∈F,有 a(f1(x)+f2(x)=f1(x0)+f2(xo)=a(f1(x))+a(f2(x) o(efi(a))=cfi()= co(i(r)) (e).是,对vf1(x),f2(x)∈F[xln和c∈F,有 a(f(x)+f2(x)=(x+a)x(f1(x)+f2(x) =(x+a)(x)+(x+a)元2(x)=(1(x)+a(2(x) d o(cfi()=(+a(cfi())=co(i(a) (f).是,对ⅤX,Y∈F"xn和Vc∈F,有 o(X+Y)=B(X+Y)C=BXC+BYC=0(X)+o(Y) o(cX)=b(eX)C= co(x) (g).是,对vf1(t),2(t)∈C0,2]和vc∈R,有 a(f1(t)+f2(t) f1(x)+f2(x)]sin(t-x)dx=a(f1(t)+a(2(t) a(cfi(t)=/cf(z) in(t-rdr=co(f(t) (h.是.(证明略)第五讲习题解答(讲义) 1. 判断下列各变换是否是线性变换. 解: (a). 是, 对∀ " x1 y1 # , " x2 y2 # ∈ R 2 和 c ∈ R, 有 σ " x1 y1 # + " x2 y2 #! = σ " x1 + x2 y1 + y2 #! = " x1 + x2 + k(y1 + y2) y1 + y2 # = " x1 + ky1 y1 # + " x2 + ky2 y2 # = σ " x1 y1 #! + σ " x2 y2 #! σ c " x1 y1 #! = σ " cx1 cy1 #! = " cx1 + k(cy1) cy1 # = c " x1 + ky1 y1 # = cσ " x1 y1 #! (b). 不是, 对 ∀x, y ∈ V , 有 σ (x + y) = a 6= a + a = σ (x) + σ (y) (c). 是, 对 ∀f1(x), f2(x) ∈ F[x]n 和 ∀c ∈ F, 有 σ (f1(x) + f2(x)) = f1(x + 1) + f2(x + 1) = σ (f1(x)) + σ (f2(x)) σ (cf1(x)) = cf1(x + 1) = cσ (f1(x)) (d). 是, 对 ∀f1(x), f2(x) ∈ F[x]n 和 ∀c ∈ F, 有 σ (f1(x) + f2(x)) = f1(x0) + f2(x0) = σ (f1(x)) + σ (f2(x)) σ (cf1(x)) = cf1(x0) = cσ (f1(x)) (e). 是, 对 ∀f1(x), f2(x) ∈ F[x]n 和 ∀c ∈ F, 有 σ (f1(x) + f2(x)) = (x + a) d dx (f1(x) + f2(x)) = (x + a) d dxf1(x) + (x + a) d dxf2(x) = σ (f1(x)) + σ (f2(x)) σ (cf1(x)) = (x + a) d dx (cf1(x)) = cσ (f1(x)) (f). 是, 对 ∀X, Y ∈ F n×n 和 ∀c ∈ F, 有 σ (X + Y) = B (X + Y) C = BXC + BYC = σ (X) + σ (Y) σ (cX) = B (cX) C = cσ (X) (g). 是, 对 ∀f1(t), f2(t) ∈ C [0, 2π] 和 ∀c ∈ R, 有 σ (f1(t) + f2(t)) = Z 2π 0 [f1(x) + f2(x)] sin (t − x) dx = σ (f1(t)) + σ (f2(t)) σ (cf1(t)) = Z 2π 0 cf1(x) sin (t − x) dx = cσ (f1(t)) (h). 是. (证明略)