Thread: Tell, if the following functions are linear functions between vector spaces.

1. 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:
ℝ³-->ℝ² f((x,y,z)) = (xy,x+y)

f(0)ofℝ³ = f(0,0,0) = (0*0, 0+0) = (0,0) = ℝ²

Showing homomorphism
v1=(x,y,z), v2=(x',y',z') ∈ ℝ³

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:
ℝ³-->ℝ³ f((x,y,z)) = (0, -x+y, 3x-5y)

f(0)ofℝ³ = f(0,0,0) = (0, 0+0, 3*0-3*0) = (0,0,0) = ℝ³

Showing homomorphism
v1=(x,y,z), v2=(x',y',z') ∈ ℝ³

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:ℝ³-->ℝ³ f((x,y,z)) = (0,0,1)

f(0)ofℝ³ = f(0,0,0) ≠ (0,0,1)
- Not a linear function between ℝ³ and ℝ³

2. 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.

3. 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)

4. Re: Tell, if the following functions are linear functions between vector spaces.

Line 3 is wrong, but otherwise, it looks good.

5. Re: Tell, if the following functions are linear functions between vector spaces.

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