Vector Space Map

Algebra Level 1

$V$ is a two-dimensional vector space .Map $f:V→R$ maps a vector to a real number .

If $∀a∈V,∀b∈V,∀λ∈R,f(λa+(1-λ)b)=λf(a)+(1-λ)f(b)$ ,let's say map $f$ has property P .

Now given these maps:

（1） $f_{1}(m)=x+y，m=(x,y)∈V$ ;

（2） $f_{2}(m)=x^2+y，m=(x,y)∈V$ ;

（3） $f_{3}(m)=x+y+1，m=(x,y)∈V$ ;

（4） $f_{4}(m)=2x+y，m=(x,y)∈V$ ;

（5） $f_{5}(m)=x，m=(x,y)∈V$ ;

Which map doesn't have property P ?

If you find the answer,please tell me the essence of the property P.

f1 f4 f3 f2 f5

1 solution

A Former Brilliant Member
Jan 19, 2018

Any able mathematician should be able to recognise that $f$ maps lines to lines, and is hence a linear map. So it really comes down to which map doesn't satisfy the linearity property. It's obvious which one doesn't, given the options.

P.S. Your $\LaTeX$ skills need a lot of work. Not a big issue, just pointing something out.

I think it isn't a traditional linear map.Since a traditional map is defined as f(a+b)=f(a)+f(b),If the property P means traditional linear map,it shows that f3 isn't linear,but actually it does satisfy the property P. I think affine map would be a better name for property P.

Alice Smith - 3 years, 4 months ago

Linear maps and affine maps only differ in that linear maps map the zero element to itself. But other than that, I agree with your last point.

A Former Brilliant Member - 3 years, 4 months ago

Vector Space Map

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