Why the intersection appears in the matrix
Its for a calculus homework. They give me 2 subspace basis and a matrix, and after row reduction it appears the sum and the intersection. I have to explain why and how appears the intersection. This is the homework.
$ basis-S = {(1,2,1,1,1),(1,0,1,0,1),(-1,1,0,1,1)} $
$ basis-W = {(1,1,1,2,-2),(1,3,1,3,-2)} $
So, in the matrix we have:
- Green -> basis-S
- Yellow -> basis-S
- Orange -> basis-W
- Purple -> null, 0
After the row reduction, we have:
- Sum -> $ (1,2,1,1,1)(0,1,0,2,-3)(0,0,-1,4,-11)(0,0,0,3,-6) $ (upper left)
- Intersection -> $ (0,2,0,1,0) $ (bottom right)
So i have to explain why appears the intersection in the right. How the information of basis-W pass to the right and appears the intersection.
If you can help me, i appreciate it. Any question, tell me. Thanks!
calculus vector-spaces matrix-calculus
asked Nov 28 '18 at 3:10
Juan Manuel
84
add a comment |
Its for a calculus homework. They give me 2 subspace basis and a matrix, and after row reduction it appears the sum and the intersection. I have to explain why and how appears the intersection. This is the homework.
$ basis-S = {(1,2,1,1,1),(1,0,1,0,1),(-1,1,0,1,1)} $
$ basis-W = {(1,1,1,2,-2),(1,3,1,3,-2)} $
So, in the matrix we have:
- Green -> basis-S
- Yellow -> basis-S
- Orange -> basis-W
- Purple -> null, 0
After the row reduction, we have:
- Sum -> $ (1,2,1,1,1)(0,1,0,2,-3)(0,0,-1,4,-11)(0,0,0,3,-6) $ (upper left)
- Intersection -> $ (0,2,0,1,0) $ (bottom right)
So i have to explain why appears the intersection in the right. How the information of basis-W pass to the right and appears the intersection.
If you can help me, i appreciate it. Any question, tell me. Thanks!
calculus vector-spaces matrix-calculus
asked Nov 28 '18 at 3:10
Juan Manuel
84
btw I haven't seen the word "scalation" before; the terms I've seen are "row reduction" or "Gaussian elimination". en.wikipedia.org/wiki/Gaussian_elimination
– stewbasic
Nov 28 '18 at 4:35
Could you give some indication which linear algebra topics you have seen? Do you know what is meant by the projection $pi_1:Voplus Vto V$ on the first factor?
– stewbasic
Nov 28 '18 at 4:41
@stewbasic sorry, english its not my first language, row reduction is what i mean. And what i have been like this V⊕V, is direct sum of subspace, i dont remember seeing projection.
– Juan Manuel
Nov 28 '18 at 11:52
add a comment |
Its for a calculus homework. They give me 2 subspace basis and a matrix, and after row reduction it appears the sum and the intersection. I have to explain why and how appears the intersection. This is the homework.
$ basis-S = {(1,2,1,1,1),(1,0,1,0,1),(-1,1,0,1,1)} $
$ basis-W = {(1,1,1,2,-2),(1,3,1,3,-2)} $
So, in the matrix we have:
- Green -> basis-S
- Yellow -> basis-S
- Orange -> basis-W
- Purple -> null, 0
After the row reduction, we have:
- Sum -> $ (1,2,1,1,1)(0,1,0,2,-3)(0,0,-1,4,-11)(0,0,0,3,-6) $ (upper left)
- Intersection -> $ (0,2,0,1,0) $ (bottom right)
So i have to explain why appears the intersection in the right. How the information of basis-W pass to the right and appears the intersection.
If you can help me, i appreciate it. Any question, tell me. Thanks!
calculus vector-spaces matrix-calculus
asked Nov 28 '18 at 3:10
Juan Manuel
84
Its for a calculus homework. They give me 2 subspace basis and a matrix, and after row reduction it appears the sum and the intersection. I have to explain why and how appears the intersection. This is the homework.
$ basis-S = {(1,2,1,1,1),(1,0,1,0,1),(-1,1,0,1,1)} $
$ basis-W = {(1,1,1,2,-2),(1,3,1,3,-2)} $
So, in the matrix we have:
- Green -> basis-S
- Yellow -> basis-S
- Orange -> basis-W
- Purple -> null, 0
After the row reduction, we have:
- Sum -> $ (1,2,1,1,1)(0,1,0,2,-3)(0,0,-1,4,-11)(0,0,0,3,-6) $ (upper left)
- Intersection -> $ (0,2,0,1,0) $ (bottom right)
So i have to explain why appears the intersection in the right. How the information of basis-W pass to the right and appears the intersection.
If you can help me, i appreciate it. Any question, tell me. Thanks!
calculus vector-spaces matrix-calculus
calculus vector-spaces matrix-calculus
asked Nov 28 '18 at 3:10
Juan Manuel
84
asked Nov 28 '18 at 3:10
Juan Manuel
84
edited Nov 28 '18 at 11:53
asked Nov 28 '18 at 3:10
Juan Manuel
84
asked Nov 28 '18 at 3:10
Juan Manuel
84
asked Nov 28 '18 at 3:10
Juan Manuel
84
84
btw I haven't seen the word "scalation" before; the terms I've seen are "row reduction" or "Gaussian elimination". en.wikipedia.org/wiki/Gaussian_elimination
– stewbasic
Nov 28 '18 at 4:35
Could you give some indication which linear algebra topics you have seen? Do you know what is meant by the projection $pi_1:Voplus Vto V$ on the first factor?
– stewbasic
Nov 28 '18 at 4:41
@stewbasic sorry, english its not my first language, row reduction is what i mean. And what i have been like this V⊕V, is direct sum of subspace, i dont remember seeing projection.
– Juan Manuel
Nov 28 '18 at 11:52
add a comment |
btw I haven't seen the word "scalation" before; the terms I've seen are "row reduction" or "Gaussian elimination". en.wikipedia.org/wiki/Gaussian_elimination
– stewbasic
Nov 28 '18 at 4:35
Could you give some indication which linear algebra topics you have seen? Do you know what is meant by the projection $pi_1:Voplus Vto V$ on the first factor?
– stewbasic
Nov 28 '18 at 4:41
@stewbasic sorry, english its not my first language, row reduction is what i mean. And what i have been like this V⊕V, is direct sum of subspace, i dont remember seeing projection.
– Juan Manuel
Nov 28 '18 at 11:52
btw I haven't seen the word "scalation" before; the terms I've seen are "row reduction" or "Gaussian elimination". en.wikipedia.org/wiki/Gaussian_elimination
– stewbasic
Nov 28 '18 at 4:35
btw I haven't seen the word "scalation" before; the terms I've seen are "row reduction" or "Gaussian elimination". en.wikipedia.org/wiki/Gaussian_elimination
– stewbasic
Nov 28 '18 at 4:35
Could you give some indication which linear algebra topics you have seen? Do you know what is meant by the projection $pi_1:Voplus Vto V$ on the first factor?
– stewbasic
Nov 28 '18 at 4:41
Could you give some indication which linear algebra topics you have seen? Do you know what is meant by the projection $pi_1:Voplus Vto V$ on the first factor?
– stewbasic
Nov 28 '18 at 4:41
@stewbasic sorry, english its not my first language, row reduction is what i mean. And what i have been like this V⊕V, is direct sum of subspace, i dont remember seeing projection.
– Juan Manuel
Nov 28 '18 at 11:52
@stewbasic sorry, english its not my first language, row reduction is what i mean. And what i have been like this V⊕V, is direct sum of subspace, i dont remember seeing projection.
– Juan Manuel
Nov 28 '18 at 11:52
add a comment |
1 Answer
1
active
oldest
votes
I'll use $B_S$ and $B_W$ to denote the matrices containing the given bases, so the initial matrix can be written in block form as
$$
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}.
$$
Hint: An element of the row space of the original matrix is of the form
$$
begin{bmatrix}a&bend{bmatrix}
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}
=begin{bmatrix}aB_S+bB_W&aB_Send{bmatrix}
$$
If the first half of this vector is zero, what does it tell you? Now recall that the row reduced matrix has the same row space as the original matrix.
answered Nov 28 '18 at 21:34
stewbasic
5,7331926
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
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oldest
votes
active
oldest
votes
I'll use $B_S$ and $B_W$ to denote the matrices containing the given bases, so the initial matrix can be written in block form as
$$
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}.
$$
Hint: An element of the row space of the original matrix is of the form
$$
begin{bmatrix}a&bend{bmatrix}
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}
=begin{bmatrix}aB_S+bB_W&aB_Send{bmatrix}
$$
If the first half of this vector is zero, what does it tell you? Now recall that the row reduced matrix has the same row space as the original matrix.
answered Nov 28 '18 at 21:34
stewbasic
5,7331926
add a comment |
I'll use $B_S$ and $B_W$ to denote the matrices containing the given bases, so the initial matrix can be written in block form as
$$
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}.
$$
Hint: An element of the row space of the original matrix is of the form
$$
begin{bmatrix}a&bend{bmatrix}
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}
=begin{bmatrix}aB_S+bB_W&aB_Send{bmatrix}
$$
If the first half of this vector is zero, what does it tell you? Now recall that the row reduced matrix has the same row space as the original matrix.
answered Nov 28 '18 at 21:34
stewbasic
5,7331926
add a comment |
I'll use $B_S$ and $B_W$ to denote the matrices containing the given bases, so the initial matrix can be written in block form as
$$
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}.
$$
Hint: An element of the row space of the original matrix is of the form
$$
begin{bmatrix}a&bend{bmatrix}
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}
=begin{bmatrix}aB_S+bB_W&aB_Send{bmatrix}
$$
If the first half of this vector is zero, what does it tell you? Now recall that the row reduced matrix has the same row space as the original matrix.
answered Nov 28 '18 at 21:34
stewbasic
5,7331926
I'll use $B_S$ and $B_W$ to denote the matrices containing the given bases, so the initial matrix can be written in block form as
$$
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}.
$$
Hint: An element of the row space of the original matrix is of the form
$$
begin{bmatrix}a&bend{bmatrix}
begin{bmatrix}B_S&B_S\B_W&0end{bmatrix}
=begin{bmatrix}aB_S+bB_W&aB_Send{bmatrix}
$$
If the first half of this vector is zero, what does it tell you? Now recall that the row reduced matrix has the same row space as the original matrix.
answered Nov 28 '18 at 21:34
stewbasic
5,7331926
answered Nov 28 '18 at 21:34
stewbasic
5,7331926
answered Nov 28 '18 at 21:34
stewbasic
5,7331926
answered Nov 28 '18 at 21:34
stewbasic
5,7331926
5,7331926
add a comment |
add a comment |
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btw I haven't seen the word "scalation" before; the terms I've seen are "row reduction" or "Gaussian elimination". en.wikipedia.org/wiki/Gaussian_elimination
– stewbasic
Nov 28 '18 at 4:35
Could you give some indication which linear algebra topics you have seen? Do you know what is meant by the projection $pi_1:Voplus Vto V$ on the first factor?
– stewbasic
Nov 28 '18 at 4:41
@stewbasic sorry, english its not my first language, row reduction is what i mean. And what i have been like this V⊕V, is direct sum of subspace, i dont remember seeing projection.
– Juan Manuel
Nov 28 '18 at 11:52