Question

Prove that \(f_1(x)=1,f_2 (x)=x,f_3 (x)=((3x^2-1))/2\) are orthogonal over \((-1,1)\) using matrices

Answer 1

@OOPS

Answer 2

I answered. :) and I am sure you got it. hihihi

Answer 3

you have deleted it before reading it completely

Answer 4

i know onlhy linear algebra, never took abstract... so im not good with fields
but i can read wiki for keywords if you show me the matrix method :)

Answer 5

It's not hard, friend, that from theoretical linear algebra, not abstract
any polynomial is in the form a + bx + cx^2 + dx^3+.......

Answer 6

okay...

Answer 7

so that {1, x, x^2, x^3,......} is the basis of polynomial set
Consider P_2, we have basis of P_2 is {1, x,x^2}

Answer 8

please wait,
so basis is a set of linear independent vectors that span the entire vector space, right ?

Answer 9

it seems like you're defining a basis whose dimension is infinite or something

Answer 10

so that the first one is 1 + 0x + 0x^2
the second is 0+1x +0x^2
the third is -1/2 +0 x+ 3/2 x^2
and Yes for your question

Answer 11

P_2 is subset of P, and Wolg P is a vector space

Answer 12

from that I can construct the matrix as shown.

Answer 13

I see...

Answer 14

And it 's quite easy to prove they are independent vectors and then orthogonal.

Answer 15

this would be the matrix \[\large \left[\begin{array}\\ 1&0&0\\0&1&0\\-1/2&0&3/2 \end{array}\right]\]

?

Answer 16

yes

Answer 17

we need to show determinant is 0 for independence

Answer 18

what about orthogonality ?

Answer 19

*not 0

Answer 20

using inner product

Answer 21

If you show that rank of the matrix =3 it should solve the problem.

Answer 22

f1 dot f2 =0 and so on...

Answer 23

orthogonal
\(AA^T =I\)

Answer 24

Sweet :3

Answer 25

this looks more elegant than the integrals xD

Answer 26

Yes, there are many ways, but I choose the simplest way for my simple brain

Answer 27

just a final noob question :
we need to take inner product of pairs of \(column\) vectors right ?

Answer 28

I use row, not column

Answer 29

It's the same but always go with the easiest.

Answer 30

row space is orthogonal <==> column space is orthogonal

?

Answer 31

The reason I choose row is (1,0,0) is the first vector as f(1), so that it makes more sense than picking (1,0,1/2) where I don't know what it stands for.

Answer 32

and I need prove f(1) perpendicular to f(2) and f(3) . That 's it

Answer 33

If you take it as a 3*3 matrix yes, you can use columns or rows.

Answer 34

Oh I see... i think i get it :) thanks !

Answer 35

:)