Question

If u =(x1,x2) , v = (y1,y2)
f(u,v) = 2x1y2-3x2y1
how to show f is a bilinear form?
Please help

Answer 1

Hi @loser! Did you check each of the sum tests?

http://en.wikipedia.org/wiki/Bilinear_form

Answer 2

I don't know what 's wrong with me. My brain is on and off with the stuff. Sometime, it's quite trivial to me but just few hours later, it disappears as if I am a dummy one. :(

Answer 3

@dan815 can you help me?

Answer 4

@wio @dumbcow

Answer 5

Prove the properties on the wikipedia page.

Answer 6

I don't know how. :(

Answer 7

Show that \[
f(u+w,v) = f(u,v) + f(w,v)
\]first

Answer 8

let \(w=\langle z_1,z_2\rangle\)

Answer 9

If you are tired, get some rest. This is too trivial for me to do it for you.

Answer 10

Please!! I spend whole day to understand the stuff. I know how to prove in symbolic form only. :(

Answer 11

I mean I can put 2 cases in 1like
f(u1,u2)(v1,v2) = f(u1, v1) + f(u1,v2)+ f(u2,v1)+ f(u2,v2)
but ...blank then

Answer 12

n \[
u+w = \langle x_1+z_1,x_2+z_2\rangle
\]

Answer 13

That's my limit.

Answer 14

Thanks anyway!!:( let me try later. So tired now

Answer 15

w=(z1,z2)
If u =(x1,x2) ,
v = (y1,y2)
f(u,v) = 2x1y2-3x2y1
f(u+v,w)=f(u,w)+f(v,w)
check if
2(x1+y1)(z2)-3(x2+y2)(z1)=(2x1z2-3x2z1)+(2y1z2-3y2z1)
2x1z2+2y1z2-3x2z1-3y2z1=(2x1z2-3x2z1)+(2y1z2-3y2z1)
(2x1z2-3x2z1)+(2y1z1-3y2z1)=(2x1z2-3x2z1)+(2y1z2-3y2z1)
so
f(u+v,w)=f(u,w)+f(v,w)
is true

Answer 16

f(u,v) = 2x1y2-3x2y1
next one
f(u ,v+w)=f(u,v)+f(u,w)
the same thing
check if
2x1(y2+z2)-3x2(y1+z1)=(2x1y2-3x2y1)+(2x1z2-3x2z1)

f(u ,v+w)=2x1(y2+z2)-3x2(y1+z1)
=2x1y2+2x1z2-3x2y1-3x2z1
=(2x1y2-3x2y1)+(2x1z2-3x2z1)
=f(u,v)+f(u,w)

so
f(u ,v+w)=f(u,v)+f(u,w)
is true

Answer 17

the last one
f(u,v) = 2x1y2-3x2y1

f(λu, v) = f(u, λv) = λf(u, v)
case 1
check
f(λu, v) = f(u, λv)
2(λx1)y2-3(λ2)y1=2x1(λy2)-3x2(λy1)

f(λu, v) = 2(λx1)y2-3(λ2)y1
=2 λx1 y2-3 λx2 y1
=2x1 λy2 -3x2 λy1
=2x1(λy2)-3x2(λy1)
= f(u, λv)


case 2
f(λu, v) = λf(u, v)
2(λx1)y2-3(λ2)y1=λ(2x1 y2 -3x2 y1)

f(λu, v) = 2(λx1)y2-3(λ2)y1
=2 λx1 y2-3 λx2 y1
=λ(2x1 y2 -3x2 y1)
=λf(u, v)


so
f(λu, v) = f(u, λv) = λf(u, v)
is true

Answer 18

Wow @ikram002p ! You are awesome!!

Answer 19

^^ np !