Lista Halliday

RESPOSTA DA 4ª LISTA

1.º Quesito

a) T: R2             R3: T(x,y) = (x-y, 3x, -2y)[pic 1]

u = (x1, y1)

v = (x2, y2)

i)        T(u+v) = T[(x1, y1) + (x2, y2)]

        T(u+v) = T[(x1+x2, y1+y2)]

        T(u+v) = [(x1+x2) - (y1+y2), 3(x1+x2), -2(y1+y2)]

        T(u+v) = (x1-y1, 3x1, -2y1) + (x2-y2, 3x2, -2y2)

        T(u+v) = T(u) + T(v)

ii)        T(αu) = T(αx1, αy1)

        T(αu) = (αx1-αy1, 3αx1, -2αy1)

        T(αu) = α(x1-y1, 3x1, -2y1)

        T(αu) = αT(u)

b) T: R3             R3: T(x,y,z) = (x+y, x-y, 0)[pic 2]

u = (x1, y1, z1)

v = (x2, y2, z2)

i)        T(u+v) = T(x1, x2, y1+y2, z1, z2)

        T(u+v) = ((x1+x2) + (y1+y2), (x1+x2) - (y1+y2), (0+0))

        T(u+v) = [(x1+y1, x1-y1, 0) + (x2+y2, x2-y2, 0)]

        T(u+v) = T(u) + T(v)

ii)        T(αu) = T(αx1, αy1, αz1)

        T(αu) = (αx1+αy1, x1-y1, α0)

        T(αu) = α( x1+y1, x1-y1, 0)

        T(αu) = αT(u)

c) T: R2             R2: T(x,y) = (x2+y2, x)[pic 3]

u = (x1, y1)

v = (x2, y2)

i)        T(u+v) = T(x1, y1) + (x2, y2)

        T(u+v) = T(x1+x2, y1+y2)

        T(u+v) = [(x1+x2)2 + (y1+y2)2, (x1+x2)]

        T(u+v) = (x12+2x1 x2+ x22+y12+2y1 y2+ y22, x1+x2)

        T(u+v) ≠ T(u) + T(v)        Não Linear

ii)        T(αu) = T(αx1, αy1)

        T(αu) = (αx12+αy12, αx1)

        T(αu) = α(x12+αy12, x1)

        T(αu) = αT(u)

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d) T: R2             R: T(x,y) = (x.y)[pic 4]

u = (x1, y1)

v = (x2, y2)

i)        T(u+v) = T[(x1, y1) + (x2, y2)]

        T(u+v) = T(x1+x2, y1+y2)

        T(u+v) = [(x1+ y1)(x2+y2)]

        T(u+v) = [(x1 x2 + x1 y2 + y1x2 + y1 y2)]

T(u+v) ≠ T(u) + T(v)        Não Linear

ii)        T(αu) = T(αx1, αy1)

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