- #1
Aleoa
- 128
- 5
HI .I'm trying to prove that, for a linear transformation, it is worth that:
[tex]f(a\bar{x}+b\bar{y})=af(\bar{x})+bf(\bar{y})[/tex] for every real numbers a and b.
Until now, I have proved by myself that
[tex]f(\bar{x}+\bar{y})=f(\bar{x})+f(\bar{y})[/tex].
and , using this result i proved that:
[tex]f(a\bar{v}) = f(\sum\bar{v})\text{(Sum it $a$ times)}[/tex].
[tex]=\sum f(\bar{v})[/tex].
[tex]=af(\bar{v})[/tex].
for every integer a.
How can I use this result in order to prove that
[tex]f(a\bar{v}) = af(\bar{v})[/tex]
for every [tex]a\in \mathbb{R} [/tex] ?
[tex]f(a\bar{x}+b\bar{y})=af(\bar{x})+bf(\bar{y})[/tex] for every real numbers a and b.
Until now, I have proved by myself that
[tex]f(\bar{x}+\bar{y})=f(\bar{x})+f(\bar{y})[/tex].
and , using this result i proved that:
[tex]f(a\bar{v}) = f(\sum\bar{v})\text{(Sum it $a$ times)}[/tex].
[tex]=\sum f(\bar{v})[/tex].
[tex]=af(\bar{v})[/tex].
for every integer a.
How can I use this result in order to prove that
[tex]f(a\bar{v}) = af(\bar{v})[/tex]
for every [tex]a\in \mathbb{R} [/tex] ?