Odwzorowania liniowe

[rafcio_100]

Które z podanych odwzorowań są liniowe?
a) \(\displaystyle{ f(x) = \left| x\right|}\)
b) \(\displaystyle{ f(x, y) = (2x + y, 3y + 6x)}\)

a) \(\displaystyle{ u_{1} = ( x_{1}, y_{1})}\)
\(\displaystyle{ u_{2} = ( x_{2}, y_{2})}\)
\(\displaystyle{ f(u_{1}+u_{2}) = f(x_{1}+x_{2}, y_{1}+y_{2}) = (|x_{1}+x_{2}|, |y_{1}+y_{2}|)}\)
\(\displaystyle{ f(u_{1})+f(u_{2}) = f(x_{1}, y_{1}) + f(x_{2}, y_{2}) = (|x_{1}|, |y_{1}|) + (|x_{2}|, |y_{2}|) = (|x_{1}| + |x_{2}|, |y_{1}| + |y_{2}|)}\)
f nie jest liniowa

b) \(\displaystyle{ u_{1} = ( x_{1}, y_{1})}\)
\(\displaystyle{ u_{2} = ( x_{2}, y_{2})}\)
\(\displaystyle{ u = ( x, y)}\)
\(\displaystyle{ f(u_{1} +u_{2}) = f(x_{1} + x_{2}, y_{1} + y_{2}) = (2x_{1} + 2x_{2} + y_{1} + y_{2}, 3y_{1} + 3y_{2} + 6x_{1} + 6x_{2}) = (2x_{1} + y_{1}, 3y_{1} + 6x_{1}) + (2x_{2} + y_{2},3y_{2} + 6x_{2} = f(x_{1},y_{1}) + f(x_{2},y_{2}) = f(u_{1}) + f(u_{2})}\)

\(\displaystyle{ f(a \cdot u) = f(ax, ay) = (2ax + ay, 3ay + 6ax) = a(2x+y,3y+6x) = a \cdot f(x,y) = a \cdot f(u)}\)
f jest liniowa

Dobrze to rozwiązałem?

[miodzio1988]

W \(\displaystyle{ a}\) masz funkcje jednej zmiennej przeciez..

[rafcio_100]

a) \(\displaystyle{ u_{1} = ( x_{1})}\)
\(\displaystyle{ u_{2} = ( x_{2})}\)
\(\displaystyle{ f(u_{1}+u_{2}) = f(x_{1}+x_{2}) = (|x_{1}+x_{2}|)}\)
\(\displaystyle{ f(u_{1})+f(u_{2}) = f(x_{1}) + f(x_{2}) = (|x_{1}|) + (|x_{2}|)}\)
f nie jest liniowa
teraz dobrze?

[akermann1]

Tak.

[yorgin]

Nie jest dobrze.

a) \(\displaystyle{ u_{1} = ( x_{1})}\)
\(\displaystyle{ u_{2} = ( x_{2})}\)
\(\displaystyle{ f(u_{1}+u_{2}) = f(x_{1}+x_{2}) = (|x_{1}+x_{2}|)}\)
\(\displaystyle{ f(u_{1})+f(u_{2}) = f(x_{1}) + f(x_{2}) = (|x_{1}|) + (|x_{2}|)}\)
f nie jest liniowa

To rozumowanie jest pozbawione jakiegokolwiek argumentu przeciw liniowości. Jest to tylko rozpisanie dwóch stron z definicji liniowiości i tyle.

Zapis \(\displaystyle{ u_1=(x_1)}\) jest również dla mnie niezrozumiały. Co to właściwie znaczy?