Question

using inner product f,g and the gram schmiddt process find an orthagonal basis for polynomial space spanned by {1,t,t^2,t^3} on (-2,2)

Answer 1

@amistre64

Answer 2

\[<f,g >=\int\limits_{a}^{b}f(t)g(t) dt \]

Answer 3

know how to find f(t) and g(t) if they are two individual functions

Answer 4

if f(t)= \[t ^{3}+t ^{2}+t +1\]

Answer 5

polys always have given me a headache in linear algebra for some reason ....

Answer 6

yes, and you want to find a g(t) such that\[\int_{-2}^{2}(t^3+t^2+t+1)\cdot g(t)~dt=0\]

Answer 7

g(t)=-t3-t2-t-1

Answer 8

well, i spose that works :)

Answer 9

but i dont understand how grahm schmidt
falls in

Answer 10

because these are not vectors.

Answer 11

in trying to refresh my memory; 2 vectors are orthogonal of their dot product is 0

we have a basis defined as: 1,t,t^2,t^3 as the column vectors of F

let G be represented by the vector: <g0,g1,g2,g3> such that

F*G = \(g_0(1)+g_1(t)+g_2(t^2)+g_3(t^3)=0\)

Answer 12

do orthogonal vectors have orthogonal tangents right? if so would we take the derivative of this setup a few times to get a system of equations?

Answer 13

that is the question

Answer 14

this seems to be trying to work the GramSchmidt with polys

http://web.gccaz.edu/~wkehowsk/225-Linear-10-11-Sp/c05s3-gram-schmidt-poly-approx-225-Sp11.pdf

Answer 15

so i just found out we are allowed to use maple for some problems

Answer 16

really .... ive never used a maple

Answer 17

but, I have come across a textbook section that is walking me thru a gram schmidt poly

Answer 18

\[f_0=1\\f_1=t\\f_2=t^2\\f_3=t^3\]

let g0=f0; g0 = 1
\[g_1=f_1-\frac{<f_1,g_0>}{||g_0||^2}g_0\]
\[g_2=f_2-\frac{<f_2,g_0>}{||g_0||^2}g_0-\frac{<f_2,g_1>}{||g_1||^2}g_1\]
etc...

Answer 19

\[g_1=1-\frac{\int_{-2}^{2}x(1)dx}{\int_{-2}^{2}1^2dx}~(1)\]
\[g_1=1\]
if im reading it correctly, page 330 round a bout

https://www.math.purdue.edu/academic/files/courses/2010spring/MA26200/4-12.pdf

Answer 20

so we get g0 to g1 and then evaluate the intergral individially so u have 4
different functions

Answer 21

correct