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Creating a matrix to project vectors in $5$-dimensional space to $3$-dimensional sub space through a point?
An orthonormal set cannot be a basis in an infinite dimension vector space?How to project a n-dimensional point onto a 2-D subspace?Projection Matrix between two VectorsQuestion about maximal orthonormal subset in infinite dimensional vector spaceUsing my orthonormal basis to find a polynomial that best approximates the polynomial $t^3$infinite dimensional hilbert space - uniqueness of series expansionCreating an orthonormal basisWhat is the maximum number of quadrants in $n$ dimensional space that a $k$ dimensional hyperplane can pass through?Is zero vector always present in any n-dimensional space?In 3D space, how to project a point into x, y and z?
$begingroup$
I am trying to self teach matrices. I am confused as to how to create a matrix to project vectors in $5$-dimensional space to a $3$-dimensional sub-space that passes through the origin and the points $(2, 0, 0, −2, 2), (0, 2, 2, 0, 0)$ and $(2, 0, 0, 2, 0)$, by using an orthonormal basis.
I understand how to project a vector $V$ onto a vector $W$ ( $fracVcdot WWcdot WW$ ), I have only used this to project $2$d vectors onto other $2$d vectors but I am not sure how to apply it to this scenario ($5$-dimension space to $3$-dimensional space).
I am also unsure as to how to project it through a certain point as given above.
Any help would be very appreciated.
linear-algebra matrices orthonormal
edited 15 hours ago
Daniele Tampieri
2,3972922
asked 16 hours ago
John A SmithJohn A Smith
1
New contributor
John A Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I am trying to self teach matrices. I am confused as to how to create a matrix to project vectors in $5$-dimensional space to a $3$-dimensional sub-space that passes through the origin and the points $(2, 0, 0, −2, 2), (0, 2, 2, 0, 0)$ and $(2, 0, 0, 2, 0)$, by using an orthonormal basis.
I understand how to project a vector $V$ onto a vector $W$ ( $fracVcdot WWcdot WW$ ), I have only used this to project $2$d vectors onto other $2$d vectors but I am not sure how to apply it to this scenario ($5$-dimension space to $3$-dimensional space).
I am also unsure as to how to project it through a certain point as given above.
Any help would be very appreciated.
linear-algebra matrices orthonormal
edited 15 hours ago
Daniele Tampieri
2,3972922
asked 16 hours ago
John A SmithJohn A Smith
1
New contributor
John A Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I am trying to self teach matrices. I am confused as to how to create a matrix to project vectors in $5$-dimensional space to a $3$-dimensional sub-space that passes through the origin and the points $(2, 0, 0, −2, 2), (0, 2, 2, 0, 0)$ and $(2, 0, 0, 2, 0)$, by using an orthonormal basis.
I understand how to project a vector $V$ onto a vector $W$ ( $fracVcdot WWcdot WW$ ), I have only used this to project $2$d vectors onto other $2$d vectors but I am not sure how to apply it to this scenario ($5$-dimension space to $3$-dimensional space).
I am also unsure as to how to project it through a certain point as given above.
Any help would be very appreciated.
linear-algebra matrices orthonormal
edited 15 hours ago
Daniele Tampieri
2,3972922
asked 16 hours ago
John A SmithJohn A Smith
1
New contributor
John A Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I am trying to self teach matrices. I am confused as to how to create a matrix to project vectors in $5$-dimensional space to a $3$-dimensional sub-space that passes through the origin and the points $(2, 0, 0, −2, 2), (0, 2, 2, 0, 0)$ and $(2, 0, 0, 2, 0)$, by using an orthonormal basis.
I understand how to project a vector $V$ onto a vector $W$ ( $fracVcdot WWcdot WW$ ), I have only used this to project $2$d vectors onto other $2$d vectors but I am not sure how to apply it to this scenario ($5$-dimension space to $3$-dimensional space).
I am also unsure as to how to project it through a certain point as given above.
Any help would be very appreciated.
linear-algebra matrices orthonormal
linear-algebra matrices orthonormal
edited 15 hours ago
Daniele Tampieri
2,3972922
asked 16 hours ago
John A SmithJohn A Smith
1
New contributor
John A Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 15 hours ago
Daniele Tampieri
2,3972922
asked 16 hours ago
John A SmithJohn A Smith
1
New contributor
John A Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 15 hours ago
Daniele Tampieri
2,3972922
edited 15 hours ago
Daniele Tampieri
2,3972922
edited 15 hours ago
Daniele Tampieri
2,3972922
2,3972922
asked 16 hours ago
John A SmithJohn A Smith
1
New contributor
John A Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 16 hours ago
John A SmithJohn A Smith
1
asked 16 hours ago
John A SmithJohn A Smith
1
1
New contributor
John A Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
John A Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
John A Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
You start by making the basis of subspace orthogonal (it will make things easier), by using, for example, Gram–Schmidt process. Lucky you, you have already an orthogonal basis, so we just divide it by 2 (to make maths simpler)
$$
v_1 = (1,0,0,-1,1),\
v_2=(0,1,1,0,0),\
v_3=(1,0,0,1,0).
$$
When you have an orthogonal basis, the projection to subspace is just the sum of projections on each basis line:
$$
P_v_1v_2v_3(u) = P_v_1(u) + P_v_2(u) + P_v_3(u) = fracucdot v_1v_1^2v_1 + fracucdot v_2v_2^2v_2+fracucdot v_3v_3^2v_3.
$$
Finally, you can use the fact that $c(acdot b)=(c otimes b )cdot a$, where $otimes$ is the tensor product:
$$
P_v_1v_2v_3(u) = left(fracv_1otimes v_1v_1^2+fracv_2otimes v_2v_2^2+fracv_3otimes v_3v_3^2right)cdot u
$$
The thing in parenthesis is your matrix of projection $P$.
$$
P=
frac13beginpmatrix1&0&0&-1&1\0&0&0&0&0\0&0&0&0&0\-1&0&0&1&-1\1&0&0&-1&1endpmatrix +
frac12beginpmatrix0&0&0&0&0\0&1&1&0&0\0&1&1&0&0\0&0&0&0&0\0&0&0&0&0endpmatrix +
frac12beginpmatrix1&0&0&1&0\0&0&0&0&0\0&0&0&0&0\1&0&0&1&0\0&0&0&0&0endpmatrix
$$
answered 15 hours ago
Vasily MitchVasily Mitch
2,3841311
$endgroup$
$begingroup$
Thanks for your reply, I am still a little bit unsure about a few things. When you are projecting the vectors v1, v2, v3 onto the vector u, what is u? Also, I haven't learnt about the tensor product, is there a way to solve this without the use of the tensor product?
$endgroup$
– John A Smith
15 hours ago
$begingroup$
I am projecting $u$ onto $v_1$. $u$ is some arbitrary vector. Tensor product is nothing fancy here, it's just a notation trick. You can write down all your components if you want.
$endgroup$
– Vasily Mitch
15 hours ago
$begingroup$
I am sorry if this sounds dumb, but what exactly is the vector u? Where are you getting it from, it doesn't seem to be an arbitrary variable as you can work out the sum of projections.
$endgroup$
– John A Smith
14 hours ago
$begingroup$
You are trying to find a projection matrix $M$, which is by the definition a matrix, that being multiplied by any vector $u$ gives you its projection $P(u)$. If you take any vector $u$, find expression for $P(u)$ and then express $P(u)=Mu$, then matrix $M$ is your projection matrix
$endgroup$
– Vasily Mitch
14 hours ago
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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votes
$begingroup$
You start by making the basis of subspace orthogonal (it will make things easier), by using, for example, Gram–Schmidt process. Lucky you, you have already an orthogonal basis, so we just divide it by 2 (to make maths simpler)
$$
v_1 = (1,0,0,-1,1),\
v_2=(0,1,1,0,0),\
v_3=(1,0,0,1,0).
$$
When you have an orthogonal basis, the projection to subspace is just the sum of projections on each basis line:
$$
P_v_1v_2v_3(u) = P_v_1(u) + P_v_2(u) + P_v_3(u) = fracucdot v_1v_1^2v_1 + fracucdot v_2v_2^2v_2+fracucdot v_3v_3^2v_3.
$$
Finally, you can use the fact that $c(acdot b)=(c otimes b )cdot a$, where $otimes$ is the tensor product:
$$
P_v_1v_2v_3(u) = left(fracv_1otimes v_1v_1^2+fracv_2otimes v_2v_2^2+fracv_3otimes v_3v_3^2right)cdot u
$$
The thing in parenthesis is your matrix of projection $P$.
$$
P=
frac13beginpmatrix1&0&0&-1&1\0&0&0&0&0\0&0&0&0&0\-1&0&0&1&-1\1&0&0&-1&1endpmatrix +
frac12beginpmatrix0&0&0&0&0\0&1&1&0&0\0&1&1&0&0\0&0&0&0&0\0&0&0&0&0endpmatrix +
frac12beginpmatrix1&0&0&1&0\0&0&0&0&0\0&0&0&0&0\1&0&0&1&0\0&0&0&0&0endpmatrix
$$
answered 15 hours ago
Vasily MitchVasily Mitch
2,3841311
$endgroup$
$begingroup$
Thanks for your reply, I am still a little bit unsure about a few things. When you are projecting the vectors v1, v2, v3 onto the vector u, what is u? Also, I haven't learnt about the tensor product, is there a way to solve this without the use of the tensor product?
$endgroup$
– John A Smith
15 hours ago
$begingroup$
I am projecting $u$ onto $v_1$. $u$ is some arbitrary vector. Tensor product is nothing fancy here, it's just a notation trick. You can write down all your components if you want.
$endgroup$
– Vasily Mitch
15 hours ago
$begingroup$
I am sorry if this sounds dumb, but what exactly is the vector u? Where are you getting it from, it doesn't seem to be an arbitrary variable as you can work out the sum of projections.
$endgroup$
– John A Smith
14 hours ago
$begingroup$
You are trying to find a projection matrix $M$, which is by the definition a matrix, that being multiplied by any vector $u$ gives you its projection $P(u)$. If you take any vector $u$, find expression for $P(u)$ and then express $P(u)=Mu$, then matrix $M$ is your projection matrix
$endgroup$
– Vasily Mitch
14 hours ago
add a comment |
$begingroup$
You start by making the basis of subspace orthogonal (it will make things easier), by using, for example, Gram–Schmidt process. Lucky you, you have already an orthogonal basis, so we just divide it by 2 (to make maths simpler)
$$
v_1 = (1,0,0,-1,1),\
v_2=(0,1,1,0,0),\
v_3=(1,0,0,1,0).
$$
When you have an orthogonal basis, the projection to subspace is just the sum of projections on each basis line:
$$
P_v_1v_2v_3(u) = P_v_1(u) + P_v_2(u) + P_v_3(u) = fracucdot v_1v_1^2v_1 + fracucdot v_2v_2^2v_2+fracucdot v_3v_3^2v_3.
$$
Finally, you can use the fact that $c(acdot b)=(c otimes b )cdot a$, where $otimes$ is the tensor product:
$$
P_v_1v_2v_3(u) = left(fracv_1otimes v_1v_1^2+fracv_2otimes v_2v_2^2+fracv_3otimes v_3v_3^2right)cdot u
$$
The thing in parenthesis is your matrix of projection $P$.
$$
P=
frac13beginpmatrix1&0&0&-1&1\0&0&0&0&0\0&0&0&0&0\-1&0&0&1&-1\1&0&0&-1&1endpmatrix +
frac12beginpmatrix0&0&0&0&0\0&1&1&0&0\0&1&1&0&0\0&0&0&0&0\0&0&0&0&0endpmatrix +
frac12beginpmatrix1&0&0&1&0\0&0&0&0&0\0&0&0&0&0\1&0&0&1&0\0&0&0&0&0endpmatrix
$$
answered 15 hours ago
Vasily MitchVasily Mitch
2,3841311
$endgroup$
$begingroup$
Thanks for your reply, I am still a little bit unsure about a few things. When you are projecting the vectors v1, v2, v3 onto the vector u, what is u? Also, I haven't learnt about the tensor product, is there a way to solve this without the use of the tensor product?
$endgroup$
– John A Smith
15 hours ago
$begingroup$
I am projecting $u$ onto $v_1$. $u$ is some arbitrary vector. Tensor product is nothing fancy here, it's just a notation trick. You can write down all your components if you want.
$endgroup$
– Vasily Mitch
15 hours ago
$begingroup$
I am sorry if this sounds dumb, but what exactly is the vector u? Where are you getting it from, it doesn't seem to be an arbitrary variable as you can work out the sum of projections.
$endgroup$
– John A Smith
14 hours ago
$begingroup$
You are trying to find a projection matrix $M$, which is by the definition a matrix, that being multiplied by any vector $u$ gives you its projection $P(u)$. If you take any vector $u$, find expression for $P(u)$ and then express $P(u)=Mu$, then matrix $M$ is your projection matrix
$endgroup$
– Vasily Mitch
14 hours ago
add a comment |
$begingroup$
You start by making the basis of subspace orthogonal (it will make things easier), by using, for example, Gram–Schmidt process. Lucky you, you have already an orthogonal basis, so we just divide it by 2 (to make maths simpler)
$$
v_1 = (1,0,0,-1,1),\
v_2=(0,1,1,0,0),\
v_3=(1,0,0,1,0).
$$
When you have an orthogonal basis, the projection to subspace is just the sum of projections on each basis line:
$$
P_v_1v_2v_3(u) = P_v_1(u) + P_v_2(u) + P_v_3(u) = fracucdot v_1v_1^2v_1 + fracucdot v_2v_2^2v_2+fracucdot v_3v_3^2v_3.
$$
Finally, you can use the fact that $c(acdot b)=(c otimes b )cdot a$, where $otimes$ is the tensor product:
$$
P_v_1v_2v_3(u) = left(fracv_1otimes v_1v_1^2+fracv_2otimes v_2v_2^2+fracv_3otimes v_3v_3^2right)cdot u
$$
The thing in parenthesis is your matrix of projection $P$.
$$
P=
frac13beginpmatrix1&0&0&-1&1\0&0&0&0&0\0&0&0&0&0\-1&0&0&1&-1\1&0&0&-1&1endpmatrix +
frac12beginpmatrix0&0&0&0&0\0&1&1&0&0\0&1&1&0&0\0&0&0&0&0\0&0&0&0&0endpmatrix +
frac12beginpmatrix1&0&0&1&0\0&0&0&0&0\0&0&0&0&0\1&0&0&1&0\0&0&0&0&0endpmatrix
$$
answered 15 hours ago
Vasily MitchVasily Mitch
2,3841311
$endgroup$
You start by making the basis of subspace orthogonal (it will make things easier), by using, for example, Gram–Schmidt process. Lucky you, you have already an orthogonal basis, so we just divide it by 2 (to make maths simpler)
$$
v_1 = (1,0,0,-1,1),\
v_2=(0,1,1,0,0),\
v_3=(1,0,0,1,0).
$$
When you have an orthogonal basis, the projection to subspace is just the sum of projections on each basis line:
$$
P_v_1v_2v_3(u) = P_v_1(u) + P_v_2(u) + P_v_3(u) = fracucdot v_1v_1^2v_1 + fracucdot v_2v_2^2v_2+fracucdot v_3v_3^2v_3.
$$
Finally, you can use the fact that $c(acdot b)=(c otimes b )cdot a$, where $otimes$ is the tensor product:
$$
P_v_1v_2v_3(u) = left(fracv_1otimes v_1v_1^2+fracv_2otimes v_2v_2^2+fracv_3otimes v_3v_3^2right)cdot u
$$
The thing in parenthesis is your matrix of projection $P$.
$$
P=
frac13beginpmatrix1&0&0&-1&1\0&0&0&0&0\0&0&0&0&0\-1&0&0&1&-1\1&0&0&-1&1endpmatrix +
frac12beginpmatrix0&0&0&0&0\0&1&1&0&0\0&1&1&0&0\0&0&0&0&0\0&0&0&0&0endpmatrix +
frac12beginpmatrix1&0&0&1&0\0&0&0&0&0\0&0&0&0&0\1&0&0&1&0\0&0&0&0&0endpmatrix
$$
answered 15 hours ago
Vasily MitchVasily Mitch
2,3841311
edited 15 hours ago
answered 15 hours ago
Vasily MitchVasily Mitch
2,3841311
answered 15 hours ago
Vasily MitchVasily Mitch
2,3841311
answered 15 hours ago
Vasily MitchVasily Mitch
2,3841311
2,3841311
$begingroup$
Thanks for your reply, I am still a little bit unsure about a few things. When you are projecting the vectors v1, v2, v3 onto the vector u, what is u? Also, I haven't learnt about the tensor product, is there a way to solve this without the use of the tensor product?
$endgroup$
– John A Smith
15 hours ago
$begingroup$
I am projecting $u$ onto $v_1$. $u$ is some arbitrary vector. Tensor product is nothing fancy here, it's just a notation trick. You can write down all your components if you want.
$endgroup$
– Vasily Mitch
15 hours ago
$begingroup$
I am sorry if this sounds dumb, but what exactly is the vector u? Where are you getting it from, it doesn't seem to be an arbitrary variable as you can work out the sum of projections.
$endgroup$
– John A Smith
14 hours ago
$begingroup$
You are trying to find a projection matrix $M$, which is by the definition a matrix, that being multiplied by any vector $u$ gives you its projection $P(u)$. If you take any vector $u$, find expression for $P(u)$ and then express $P(u)=Mu$, then matrix $M$ is your projection matrix
$endgroup$
– Vasily Mitch
14 hours ago
add a comment |
$begingroup$
Thanks for your reply, I am still a little bit unsure about a few things. When you are projecting the vectors v1, v2, v3 onto the vector u, what is u? Also, I haven't learnt about the tensor product, is there a way to solve this without the use of the tensor product?
$endgroup$
– John A Smith
15 hours ago
$begingroup$
I am projecting $u$ onto $v_1$. $u$ is some arbitrary vector. Tensor product is nothing fancy here, it's just a notation trick. You can write down all your components if you want.
$endgroup$
– Vasily Mitch
15 hours ago
$begingroup$
I am sorry if this sounds dumb, but what exactly is the vector u? Where are you getting it from, it doesn't seem to be an arbitrary variable as you can work out the sum of projections.
$endgroup$
– John A Smith
14 hours ago
$begingroup$
You are trying to find a projection matrix $M$, which is by the definition a matrix, that being multiplied by any vector $u$ gives you its projection $P(u)$. If you take any vector $u$, find expression for $P(u)$ and then express $P(u)=Mu$, then matrix $M$ is your projection matrix
$endgroup$
– Vasily Mitch
14 hours ago
$begingroup$
Thanks for your reply, I am still a little bit unsure about a few things. When you are projecting the vectors v1, v2, v3 onto the vector u, what is u? Also, I haven't learnt about the tensor product, is there a way to solve this without the use of the tensor product?
$endgroup$
– John A Smith
15 hours ago
$begingroup$
Thanks for your reply, I am still a little bit unsure about a few things. When you are projecting the vectors v1, v2, v3 onto the vector u, what is u? Also, I haven't learnt about the tensor product, is there a way to solve this without the use of the tensor product?
$endgroup$
– John A Smith
15 hours ago
$begingroup$
I am projecting $u$ onto $v_1$. $u$ is some arbitrary vector. Tensor product is nothing fancy here, it's just a notation trick. You can write down all your components if you want.
$endgroup$
– Vasily Mitch
15 hours ago
$begingroup$
I am projecting $u$ onto $v_1$. $u$ is some arbitrary vector. Tensor product is nothing fancy here, it's just a notation trick. You can write down all your components if you want.
$endgroup$
– Vasily Mitch
15 hours ago
$begingroup$
I am sorry if this sounds dumb, but what exactly is the vector u? Where are you getting it from, it doesn't seem to be an arbitrary variable as you can work out the sum of projections.
$endgroup$
– John A Smith
14 hours ago
$begingroup$
I am sorry if this sounds dumb, but what exactly is the vector u? Where are you getting it from, it doesn't seem to be an arbitrary variable as you can work out the sum of projections.
$endgroup$
– John A Smith
14 hours ago
$begingroup$
You are trying to find a projection matrix $M$, which is by the definition a matrix, that being multiplied by any vector $u$ gives you its projection $P(u)$. If you take any vector $u$, find expression for $P(u)$ and then express $P(u)=Mu$, then matrix $M$ is your projection matrix
$endgroup$
– Vasily Mitch
14 hours ago
$begingroup$
You are trying to find a projection matrix $M$, which is by the definition a matrix, that being multiplied by any vector $u$ gives you its projection $P(u)$. If you take any vector $u$, find expression for $P(u)$ and then express $P(u)=Mu$, then matrix $M$ is your projection matrix
$endgroup$
– Vasily Mitch
14 hours ago
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