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Math Help - Tell, if the following functions are linear functions between vector spaces.

  1. #1
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    Tell, if the following functions are linear functions between vector spaces.

    My problem is, that my mathematical understanding is full of gaps and I am stuck at some steps or I dont know if I am finished or not.
    Sorry for the text.

    a)f:
    ℝ-->ℝ f((x,y,z)) = (xy,x+y)

    f(0)ofℝ = f(0,0,0) = (0*0, 0+0) = (0,0) = ℝ

    Showing homomorphism
    v1=(x,y,z), v2=(x',y',z') ∈

    Left side:
    f((x,y,z)+(x',y',z')) = f(x+x', y+y', z+z') = ((x+x')*(y+y'), (x+x')+(y+y'))

    Right side:
    f(x,y,z)+f(x',y',z') = (xy, x+y) + (x'y', x'+y') = ((xy+x'y'), (x+y+x'+y'))

    Am I done here or can I transform the term to make both parts equal?

    b)f:
    ℝ-->ℝ f((x,y,z)) = (0, -x+y, 3x-5y)

    f(0)ofℝ = f(0,0,0) = (0, 0+0, 3*0-3*0) = (0,0,0) = ℝ

    Showing homomorphism
    v1=(x,y,z), v2=(x',y',z') ∈

    Left side:
    f((x,y,z)+(x',y',z')) = f(x+x', y+y', z+z') = (0 , ((-x+(-x'))+(y+y')) , (3*(x+x')-5*(y+y')))

    Right side:
    f(x,y,z)+f(x',y',z') = (0 , -x+y , 3x-5y) + (0 , -x'+y' , 3x'-5y') = (0 , ((-x+(-x'))+(y+y')) , (3*(x+x')-5*(y+y')))

    Its the same as far as I see it. It this correct and can I continue the proof now?

    c)f:ℝ-->ℝ f((x,y,z)) = (0,0,1)

    f(0)ofℝ = f(0,0,0) ≠ (0,0,1)
    - Not a linear function between ℝ and ℝ
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  2. #2
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    Re: Tell, if the following functions are linear functions between vector spaces.

    a) You can foil out the LHS to show that they are NOT equal. The function is NOT a homomorphism.

    b) Yes, now you need to show that f(cv) = cf(v).

    c) Correct.
    Thanks from Cyganek
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  3. #3
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    Re: Tell, if the following functions are linear functions between vector spaces.

    for b) Is this correct? I think I messed up somewhere in the middle:

    f(c*(x,y,z))
    = f(cx,cy,cz)
    = (0, cx, cy)
    = (0 , (c(-x+y)) , (c(3x-5y))
    = c*(0 , (-x+y) , (3x-5y)
    = cf(x,y,z)
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  4. #4
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    Re: Tell, if the following functions are linear functions between vector spaces.

    Line 3 is wrong, but otherwise, it looks good.
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  5. #5
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    Re: Tell, if the following functions are linear functions between vector spaces.

    Can you tell me what exactly and how its correct please?
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