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Use Gram-Schmidt orthogonalisation to orthogonalise the system of vectors
Clash Royale CLAN TAG#URR8PPP
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I have been working on this problem, we are given the below system of vectors
$f_1 = x, f_2 = cos(x), f_3= sin(x)$ from the inner product of $C_mathbbR[-1,1]$
and we have to orthogonalise the system
I know that the Gram-Schmidt algorithm works like below
$w_1 = v_1$
$w_2 = v_2 - operatornameproj_w_1(v_2)$
and so on, and I am just struggling how to use all of this to orthgonalise the vectors. Do I have to integrate over the inner product space? Or am I just simply substituting into the Gram- Schmidt Algorithm?
Any help would be much appreciated
calculus linear-algebra vectors orthogonality orthogonal-polynomials
asked yesterday
Tom Heeley
797
add a comment |Â
up vote
1
down vote
favorite
I have been working on this problem, we are given the below system of vectors
$f_1 = x, f_2 = cos(x), f_3= sin(x)$ from the inner product of $C_mathbbR[-1,1]$
and we have to orthogonalise the system
I know that the Gram-Schmidt algorithm works like below
$w_1 = v_1$
$w_2 = v_2 - operatornameproj_w_1(v_2)$
and so on, and I am just struggling how to use all of this to orthgonalise the vectors. Do I have to integrate over the inner product space? Or am I just simply substituting into the Gram- Schmidt Algorithm?
Any help would be much appreciated
calculus linear-algebra vectors orthogonality orthogonal-polynomials
asked yesterday
Tom Heeley
797
You need to use the inner product. Both for projection and for normalization
– Andrei
yesterday
How do I do that?
– Tom Heeley
yesterday
The projection in an inner product space could be written explicitly using inner products. References could be linear algebra textbooks.
– xbh
yesterday
Start by normalizing $f_1$: $w_1=fracf_1$, where $||f||=int_-1^1f^2(x)dx$
– Andrei
yesterday
Which inner product of $C_BbbR[-1,1]$ are you talking about? There are several! The simplest is $langle f,grangle=int_-1^1 f(x)g(x),dx$, but you can also have a positive definite weight function $w(x)$ as a third factor in the integrand. This observation actually leads up to an answer of sorts. The outcome depends on the choice of the inner product so, yes, you absolutely must calculate those inner products. No other way.
– Jyrki Lahtonen
yesterday
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have been working on this problem, we are given the below system of vectors
$f_1 = x, f_2 = cos(x), f_3= sin(x)$ from the inner product of $C_mathbbR[-1,1]$
and we have to orthogonalise the system
I know that the Gram-Schmidt algorithm works like below
$w_1 = v_1$
$w_2 = v_2 - operatornameproj_w_1(v_2)$
and so on, and I am just struggling how to use all of this to orthgonalise the vectors. Do I have to integrate over the inner product space? Or am I just simply substituting into the Gram- Schmidt Algorithm?
Any help would be much appreciated
calculus linear-algebra vectors orthogonality orthogonal-polynomials
asked yesterday
Tom Heeley
797
I have been working on this problem, we are given the below system of vectors
$f_1 = x, f_2 = cos(x), f_3= sin(x)$ from the inner product of $C_mathbbR[-1,1]$
and we have to orthogonalise the system
I know that the Gram-Schmidt algorithm works like below
$w_1 = v_1$
$w_2 = v_2 - operatornameproj_w_1(v_2)$
and so on, and I am just struggling how to use all of this to orthgonalise the vectors. Do I have to integrate over the inner product space? Or am I just simply substituting into the Gram- Schmidt Algorithm?
Any help would be much appreciated
calculus linear-algebra vectors orthogonality orthogonal-polynomials
asked yesterday
Tom Heeley
797
edited yesterday
asked yesterday
Tom Heeley
797
asked yesterday
Tom Heeley
797
asked yesterday
Tom Heeley
797
797
You need to use the inner product. Both for projection and for normalization
– Andrei
yesterday
How do I do that?
– Tom Heeley
yesterday
The projection in an inner product space could be written explicitly using inner products. References could be linear algebra textbooks.
– xbh
yesterday
Start by normalizing $f_1$: $w_1=fracf_1$, where $||f||=int_-1^1f^2(x)dx$
– Andrei
yesterday
Which inner product of $C_BbbR[-1,1]$ are you talking about? There are several! The simplest is $langle f,grangle=int_-1^1 f(x)g(x),dx$, but you can also have a positive definite weight function $w(x)$ as a third factor in the integrand. This observation actually leads up to an answer of sorts. The outcome depends on the choice of the inner product so, yes, you absolutely must calculate those inner products. No other way.
– Jyrki Lahtonen
yesterday
add a comment |Â
You need to use the inner product. Both for projection and for normalization
– Andrei
yesterday
How do I do that?
– Tom Heeley
yesterday
The projection in an inner product space could be written explicitly using inner products. References could be linear algebra textbooks.
– xbh
yesterday
Start by normalizing $f_1$: $w_1=fracf_1$, where $||f||=int_-1^1f^2(x)dx$
– Andrei
yesterday
Which inner product of $C_BbbR[-1,1]$ are you talking about? There are several! The simplest is $langle f,grangle=int_-1^1 f(x)g(x),dx$, but you can also have a positive definite weight function $w(x)$ as a third factor in the integrand. This observation actually leads up to an answer of sorts. The outcome depends on the choice of the inner product so, yes, you absolutely must calculate those inner products. No other way.
– Jyrki Lahtonen
yesterday
You need to use the inner product. Both for projection and for normalization
– Andrei
yesterday
You need to use the inner product. Both for projection and for normalization
– Andrei
yesterday
How do I do that?
– Tom Heeley
yesterday
How do I do that?
– Tom Heeley
yesterday
The projection in an inner product space could be written explicitly using inner products. References could be linear algebra textbooks.
– xbh
yesterday
The projection in an inner product space could be written explicitly using inner products. References could be linear algebra textbooks.
– xbh
yesterday
Start by normalizing $f_1$: $w_1=fracf_1$, where $||f||=int_-1^1f^2(x)dx$
– Andrei
yesterday
Start by normalizing $f_1$: $w_1=fracf_1$, where $||f||=int_-1^1f^2(x)dx$
– Andrei
yesterday
Which inner product of $C_BbbR[-1,1]$ are you talking about? There are several! The simplest is $langle f,grangle=int_-1^1 f(x)g(x),dx$, but you can also have a positive definite weight function $w(x)$ as a third factor in the integrand. This observation actually leads up to an answer of sorts. The outcome depends on the choice of the inner product so, yes, you absolutely must calculate those inner products. No other way.
– Jyrki Lahtonen
yesterday
Which inner product of $C_BbbR[-1,1]$ are you talking about? There are several! The simplest is $langle f,grangle=int_-1^1 f(x)g(x),dx$, but you can also have a positive definite weight function $w(x)$ as a third factor in the integrand. This observation actually leads up to an answer of sorts. The outcome depends on the choice of the inner product so, yes, you absolutely must calculate those inner products. No other way.
– Jyrki Lahtonen
yesterday
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
Do I have to integrate over the inner product space?
The vector space of real functions whose domain is an closed interval [a,b] with inner product $langle f_1, f_2 rangle $ is $int_a^b f_1 f_2,dx$. So yes, you need to integrate over the inner product space.
You need to start by normalizing $f_1$. Let's say $w_1$ is $f_1$ normalized.
$$
w_1 = fracf_1, ||f_1||^2=int_a^bf_1^2(x)dx qquad(1)
$$
After normalizing $f_1$, you need to find a vector orthogonal to $w_1$. We can use $f_2$ to find this vector $W_2$ as
$$
W_2 = f_2-bigl(;langle, w_1,f_2,rangle;bigr)w_1;
$$
where $langle, w_1,f_2,rangle$ is the inner product of $w_1$ and $f_2$
$$
W_2 = sin x - Bigl(int_a^bw_1sin x,dxBigr)w_1 qquad(2)
$$
$W_2$ is not normalized and it can be normalized into $w_2$ in the same way as in eq(1).
Next $W_3$ and $w_3$ is found in the same way as in eq(2), which will give you 3 orthonormal vectors
answered yesterday
artha
264
See the edits I made in the source for a few tips about using TeX-spacings as well as inserting regular text inside a piece of displayed math. Mind you, I'm not sure it is a good idea to have that sentence where $langle w_1,f_2rangle$ ... on the displayed line as opposed to as regular text on the next line, but as you wanted to do it that way, I wanted to show how to do it right.
– Jyrki Lahtonen
yesterday
Thank you for the edit. I am new to the TeX so I will need this guidance from time to time. And yeah I think you are right about the "where... " sentence, I will edit that to a new line
– artha
yesterday
And regarding the inner product space, $C_BbbR[-1,1]$, Its difficult to integrate sinx and cosx over this range. If the OP's question is from a text, I doubt the author would have given this interval to solve the question. May be OP has got the range wrong. I don't have enough rep to comment on OP's, so instead of solving for this range, I just assumed [a,b].
– artha
yesterday
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Do I have to integrate over the inner product space?
The vector space of real functions whose domain is an closed interval [a,b] with inner product $langle f_1, f_2 rangle $ is $int_a^b f_1 f_2,dx$. So yes, you need to integrate over the inner product space.
You need to start by normalizing $f_1$. Let's say $w_1$ is $f_1$ normalized.
$$
w_1 = fracf_1, ||f_1||^2=int_a^bf_1^2(x)dx qquad(1)
$$
After normalizing $f_1$, you need to find a vector orthogonal to $w_1$. We can use $f_2$ to find this vector $W_2$ as
$$
W_2 = f_2-bigl(;langle, w_1,f_2,rangle;bigr)w_1;
$$
where $langle, w_1,f_2,rangle$ is the inner product of $w_1$ and $f_2$
$$
W_2 = sin x - Bigl(int_a^bw_1sin x,dxBigr)w_1 qquad(2)
$$
$W_2$ is not normalized and it can be normalized into $w_2$ in the same way as in eq(1).
Next $W_3$ and $w_3$ is found in the same way as in eq(2), which will give you 3 orthonormal vectors
answered yesterday
artha
264
See the edits I made in the source for a few tips about using TeX-spacings as well as inserting regular text inside a piece of displayed math. Mind you, I'm not sure it is a good idea to have that sentence where $langle w_1,f_2rangle$ ... on the displayed line as opposed to as regular text on the next line, but as you wanted to do it that way, I wanted to show how to do it right.
– Jyrki Lahtonen
yesterday
Thank you for the edit. I am new to the TeX so I will need this guidance from time to time. And yeah I think you are right about the "where... " sentence, I will edit that to a new line
– artha
yesterday
And regarding the inner product space, $C_BbbR[-1,1]$, Its difficult to integrate sinx and cosx over this range. If the OP's question is from a text, I doubt the author would have given this interval to solve the question. May be OP has got the range wrong. I don't have enough rep to comment on OP's, so instead of solving for this range, I just assumed [a,b].
– artha
yesterday
add a comment |Â
up vote
0
down vote
Do I have to integrate over the inner product space?
The vector space of real functions whose domain is an closed interval [a,b] with inner product $langle f_1, f_2 rangle $ is $int_a^b f_1 f_2,dx$. So yes, you need to integrate over the inner product space.
You need to start by normalizing $f_1$. Let's say $w_1$ is $f_1$ normalized.
$$
w_1 = fracf_1, ||f_1||^2=int_a^bf_1^2(x)dx qquad(1)
$$
After normalizing $f_1$, you need to find a vector orthogonal to $w_1$. We can use $f_2$ to find this vector $W_2$ as
$$
W_2 = f_2-bigl(;langle, w_1,f_2,rangle;bigr)w_1;
$$
where $langle, w_1,f_2,rangle$ is the inner product of $w_1$ and $f_2$
$$
W_2 = sin x - Bigl(int_a^bw_1sin x,dxBigr)w_1 qquad(2)
$$
$W_2$ is not normalized and it can be normalized into $w_2$ in the same way as in eq(1).
Next $W_3$ and $w_3$ is found in the same way as in eq(2), which will give you 3 orthonormal vectors
answered yesterday
artha
264
See the edits I made in the source for a few tips about using TeX-spacings as well as inserting regular text inside a piece of displayed math. Mind you, I'm not sure it is a good idea to have that sentence where $langle w_1,f_2rangle$ ... on the displayed line as opposed to as regular text on the next line, but as you wanted to do it that way, I wanted to show how to do it right.
– Jyrki Lahtonen
yesterday
Thank you for the edit. I am new to the TeX so I will need this guidance from time to time. And yeah I think you are right about the "where... " sentence, I will edit that to a new line
– artha
yesterday
And regarding the inner product space, $C_BbbR[-1,1]$, Its difficult to integrate sinx and cosx over this range. If the OP's question is from a text, I doubt the author would have given this interval to solve the question. May be OP has got the range wrong. I don't have enough rep to comment on OP's, so instead of solving for this range, I just assumed [a,b].
– artha
yesterday
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Do I have to integrate over the inner product space?
The vector space of real functions whose domain is an closed interval [a,b] with inner product $langle f_1, f_2 rangle $ is $int_a^b f_1 f_2,dx$. So yes, you need to integrate over the inner product space.
You need to start by normalizing $f_1$. Let's say $w_1$ is $f_1$ normalized.
$$
w_1 = fracf_1, ||f_1||^2=int_a^bf_1^2(x)dx qquad(1)
$$
After normalizing $f_1$, you need to find a vector orthogonal to $w_1$. We can use $f_2$ to find this vector $W_2$ as
$$
W_2 = f_2-bigl(;langle, w_1,f_2,rangle;bigr)w_1;
$$
where $langle, w_1,f_2,rangle$ is the inner product of $w_1$ and $f_2$
$$
W_2 = sin x - Bigl(int_a^bw_1sin x,dxBigr)w_1 qquad(2)
$$
$W_2$ is not normalized and it can be normalized into $w_2$ in the same way as in eq(1).
Next $W_3$ and $w_3$ is found in the same way as in eq(2), which will give you 3 orthonormal vectors
answered yesterday
artha
264
Do I have to integrate over the inner product space?
The vector space of real functions whose domain is an closed interval [a,b] with inner product $langle f_1, f_2 rangle $ is $int_a^b f_1 f_2,dx$. So yes, you need to integrate over the inner product space.
You need to start by normalizing $f_1$. Let's say $w_1$ is $f_1$ normalized.
$$
w_1 = fracf_1, ||f_1||^2=int_a^bf_1^2(x)dx qquad(1)
$$
After normalizing $f_1$, you need to find a vector orthogonal to $w_1$. We can use $f_2$ to find this vector $W_2$ as
$$
W_2 = f_2-bigl(;langle, w_1,f_2,rangle;bigr)w_1;
$$
where $langle, w_1,f_2,rangle$ is the inner product of $w_1$ and $f_2$
$$
W_2 = sin x - Bigl(int_a^bw_1sin x,dxBigr)w_1 qquad(2)
$$
$W_2$ is not normalized and it can be normalized into $w_2$ in the same way as in eq(1).
Next $W_3$ and $w_3$ is found in the same way as in eq(2), which will give you 3 orthonormal vectors
answered yesterday
artha
264
edited yesterday
answered yesterday
artha
264
answered yesterday
artha
264
answered yesterday
artha
264
264
See the edits I made in the source for a few tips about using TeX-spacings as well as inserting regular text inside a piece of displayed math. Mind you, I'm not sure it is a good idea to have that sentence where $langle w_1,f_2rangle$ ... on the displayed line as opposed to as regular text on the next line, but as you wanted to do it that way, I wanted to show how to do it right.
– Jyrki Lahtonen
yesterday
Thank you for the edit. I am new to the TeX so I will need this guidance from time to time. And yeah I think you are right about the "where... " sentence, I will edit that to a new line
– artha
yesterday
And regarding the inner product space, $C_BbbR[-1,1]$, Its difficult to integrate sinx and cosx over this range. If the OP's question is from a text, I doubt the author would have given this interval to solve the question. May be OP has got the range wrong. I don't have enough rep to comment on OP's, so instead of solving for this range, I just assumed [a,b].
– artha
yesterday
add a comment |Â
See the edits I made in the source for a few tips about using TeX-spacings as well as inserting regular text inside a piece of displayed math. Mind you, I'm not sure it is a good idea to have that sentence where $langle w_1,f_2rangle$ ... on the displayed line as opposed to as regular text on the next line, but as you wanted to do it that way, I wanted to show how to do it right.
– Jyrki Lahtonen
yesterday
Thank you for the edit. I am new to the TeX so I will need this guidance from time to time. And yeah I think you are right about the "where... " sentence, I will edit that to a new line
– artha
yesterday
And regarding the inner product space, $C_BbbR[-1,1]$, Its difficult to integrate sinx and cosx over this range. If the OP's question is from a text, I doubt the author would have given this interval to solve the question. May be OP has got the range wrong. I don't have enough rep to comment on OP's, so instead of solving for this range, I just assumed [a,b].
– artha
yesterday
See the edits I made in the source for a few tips about using TeX-spacings as well as inserting regular text inside a piece of displayed math. Mind you, I'm not sure it is a good idea to have that sentence where $langle w_1,f_2rangle$ ... on the displayed line as opposed to as regular text on the next line, but as you wanted to do it that way, I wanted to show how to do it right.
– Jyrki Lahtonen
yesterday
See the edits I made in the source for a few tips about using TeX-spacings as well as inserting regular text inside a piece of displayed math. Mind you, I'm not sure it is a good idea to have that sentence where $langle w_1,f_2rangle$ ... on the displayed line as opposed to as regular text on the next line, but as you wanted to do it that way, I wanted to show how to do it right.
– Jyrki Lahtonen
yesterday
Thank you for the edit. I am new to the TeX so I will need this guidance from time to time. And yeah I think you are right about the "where... " sentence, I will edit that to a new line
– artha
yesterday
Thank you for the edit. I am new to the TeX so I will need this guidance from time to time. And yeah I think you are right about the "where... " sentence, I will edit that to a new line
– artha
yesterday
And regarding the inner product space, $C_BbbR[-1,1]$, Its difficult to integrate sinx and cosx over this range. If the OP's question is from a text, I doubt the author would have given this interval to solve the question. May be OP has got the range wrong. I don't have enough rep to comment on OP's, so instead of solving for this range, I just assumed [a,b].
– artha
yesterday
And regarding the inner product space, $C_BbbR[-1,1]$, Its difficult to integrate sinx and cosx over this range. If the OP's question is from a text, I doubt the author would have given this interval to solve the question. May be OP has got the range wrong. I don't have enough rep to comment on OP's, so instead of solving for this range, I just assumed [a,b].
– artha
yesterday
add a comment |Â
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You need to use the inner product. Both for projection and for normalization
– Andrei
yesterday
How do I do that?
– Tom Heeley
yesterday
The projection in an inner product space could be written explicitly using inner products. References could be linear algebra textbooks.
– xbh
yesterday
Start by normalizing $f_1$: $w_1=fracf_1$, where $||f||=int_-1^1f^2(x)dx$
– Andrei
yesterday
Which inner product of $C_BbbR[-1,1]$ are you talking about? There are several! The simplest is $langle f,grangle=int_-1^1 f(x)g(x),dx$, but you can also have a positive definite weight function $w(x)$ as a third factor in the integrand. This observation actually leads up to an answer of sorts. The outcome depends on the choice of the inner product so, yes, you absolutely must calculate those inner products. No other way.
– Jyrki Lahtonen
yesterday