I need help with question 1, 3 and 4.
#Orthogonality and projection problems
stoic hedge
wide perchBOT
I need help with question 1, 3 and 4.
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stoic hedge
for question 3 since ik the basis of P1 is {1,x}, do i just plug what in to my integral to get a 2x2 matrix?
if yes, what do i do after that?
mental island
for question 3 since ik the basis of P1 is {1,x}, do i just plug what in to my i...
just use the projection formula for 3
Given an orthonormal basis ${u_1,...,u_n}$ of $U\subseteq V$, the unique projection $P\colon V\to V$ such that $P(V)=U$ is given by $P[v]=\sum_{i=1}^n\langle v,u_i\rangle u_i$
dusty drumBOT
Omegabet_
mental island
per the hint, convert ${1,x}$ into an orthonormal basis, then just compute the projection $P[e^x]$
dusty drumBOT
Omegabet_
autumn ridge
1,4 are routine checks, follow omega's advice on 3
stoic hedge
how do i apply gram Schmidt process to {1,x}?
autumn ridge
are they orthogonal to start with?
(and what does orthogonality mean)
@stoic hedge
autumn ridge
yes, orthogonal if and only if dot product is 0
stoic hedge
Yes so I don’t need to apply gram Schmidt process on it if it’s already orthogonal
I just need to make them orthonormal
dusty drumBOT
aL
stoic hedge
why am i calculating the integral?
stoic hedge
omg
autumn ridge
it's not just arbitrary information in the text, you have to use it
stoic hedge
They have given a new definition for the dot product got it
stoic hedge
I was under the assumption that I’m gonna use the basic formula
Ok I understand this one now
autumn ridge
so, tl;dr the vectors are not orthogonal, hence apply gram schmidt to orthogonalise the system
stoic hedge
Ok ok
autumn ridge
I was under the assumption that I’m gonna use the basic formula
which basic formula? 
autumn ridge
given two functions f(x) = 1 and g(x) = x, what are a1,a2 and so on?
autumn ridge
so how were you going to calculate their dot product? 😄
stoic hedge
Literally just multiply them and equal them to 0
Which is why I was saying that I’m just gonna get x = 0 which makes not sense
No*
autumn ridge
**Omegabet\_**
4.3 follows this idea
stoic hedge
Ok I just have 1 more question
autumn ridge
you should have way more
stoic hedge
In 3, when I make the set orthonormal
How do I use the projection formula for e^x
I’ve only ever used it for vectors
autumn ridge
you integrate
e^x is an element of C[0,1]
and you have the subspace {1,x} with respect to an orthonormal basis {f_1,f_2}
then you calculate <e^x, f_1> and <e^x,f_2>
autumn ridge
but you already demonstrated that you have conceptional misunderstandings, so let's make sure you get this done correctly
autumn ridge
no rush, write an update when you get back to it
autumn ridge
no worries
stoic hedge
For taking time to really go step by step
autumn ridge
I think we have ways to go still
stoic hedge
My knowledge of linear algebra has major holes in it but my goal is to study it just enough to get a good grade
The last math course that I really took interest in was differential equations
This course is a bit too abstract for me personally
autumn ridge
you don't have to have a deep understanding of linear algebra, but we would at least make sure your technique is correct
stoic hedge
True
stoic hedge
@autumn ridge
Why is the norm of x not underoot 2?
Does the formula of norm change with inner project?
Nvm I googled it
mental island
Does the formula of norm change with inner project?
Given an inner product, the induced norm is $\norm{x}:=\sqrt{\langle x,x\rangle}$. This is always true, what changes is the IP itself
dusty drumBOT
Omegabet_
stoic hedge
@autumn ridge ok im back
when calculating the new vectors using the inner product
both my calculated vectors are same
stoic hedge
ok nmv got it
stoic hedge
ok i did it
im sure my linear algebra is correct in the solution
if anything its my integration that might cause calculation mistakes
autumn ridge
how do you apply gram schmidt to 1,x
autumn ridge
yeah that's fine
stoic hedge
but its basically <x,1>/<1,1> (1)
autumn ridge
$$ f_1 = 1\quad\mbox{and}\quad f_2 = x + \alpha f_1 $$
dusty drumBOT
aL
autumn ridge
and you determine alpha from the orthogonality condition
$$ \int _0^1 (x+\alpha)dx = 0 $$
dusty drumBOT
aL
stoic hedge
oh shit actually i made a mistak
i did not make the vectors orthonormal
to make x - 1/2 orthonormal i just do (x - 1/2) / (<x - 1/2, x - 1/2>)?
autumn ridge
you are dividing by norm squared atm
autumn ridge
divide by norm