What exactly is a unique subspace?
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I am trying to solve a problem that is asking for what conditions that make $<A>$ unique. It does not explain what <> represents, but I think it means the column space. A is a matrix where its columns are a subset of the k largest eigenvectors of B.
I do not understand what it means for the column space to be unique. But if I had to guess, I would take it to mean that A is full rank, i.e., the subset of eigenvectors that form A are linearly independent.
Does anyone have a different interpretation?
Edit: Actually I just confirmed that $<A>$ is the column space of A.
matrices eigenvalues-eigenvectors independence matrix-rank
asked Nov 20 at 21:34
Iamanon
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up vote
0
down vote
favorite
I am trying to solve a problem that is asking for what conditions that make $<A>$ unique. It does not explain what <> represents, but I think it means the column space. A is a matrix where its columns are a subset of the k largest eigenvectors of B.
I do not understand what it means for the column space to be unique. But if I had to guess, I would take it to mean that A is full rank, i.e., the subset of eigenvectors that form A are linearly independent.
Does anyone have a different interpretation?
Edit: Actually I just confirmed that $<A>$ is the column space of A.
matrices eigenvalues-eigenvectors independence matrix-rank
asked Nov 20 at 21:34
Iamanon
1157
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to solve a problem that is asking for what conditions that make $<A>$ unique. It does not explain what <> represents, but I think it means the column space. A is a matrix where its columns are a subset of the k largest eigenvectors of B.
I do not understand what it means for the column space to be unique. But if I had to guess, I would take it to mean that A is full rank, i.e., the subset of eigenvectors that form A are linearly independent.
Does anyone have a different interpretation?
Edit: Actually I just confirmed that $<A>$ is the column space of A.
matrices eigenvalues-eigenvectors independence matrix-rank
asked Nov 20 at 21:34
Iamanon
1157
I am trying to solve a problem that is asking for what conditions that make $<A>$ unique. It does not explain what <> represents, but I think it means the column space. A is a matrix where its columns are a subset of the k largest eigenvectors of B.
I do not understand what it means for the column space to be unique. But if I had to guess, I would take it to mean that A is full rank, i.e., the subset of eigenvectors that form A are linearly independent.
Does anyone have a different interpretation?
Edit: Actually I just confirmed that $<A>$ is the column space of A.
matrices eigenvalues-eigenvectors independence matrix-rank
matrices eigenvalues-eigenvectors independence matrix-rank
asked Nov 20 at 21:34
Iamanon
1157
asked Nov 20 at 21:34
Iamanon
1157
asked Nov 20 at 21:34
Iamanon
1157
asked Nov 20 at 21:34
Iamanon
1157
asked Nov 20 at 21:34
Iamanon
1157
1157
add a comment |
add a comment |
1 Answer
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The column space is unique as it is defined as the smallest (wrt inclusion) subspace that contains all the columns of $A$ as elements. In this case it also is the set of all linear combinations of columns of $A$, which is minimal (as we can check it is a subspace and any subspace containing the columns must contain their linear combinations).
answered Nov 20 at 21:43
Henno Brandsma
103k346113
Thanks. So if $A$ is full rank, then the column space of A is unique?
– Iamanon
Nov 20 at 21:44
@Iamanon It's always unique, but with full rank it is equal to the maximal subspace (the whole vector space the columns lie in), so $mathbb{R}^n$ in the case of a real-valued $n times m$ matrix.
– Henno Brandsma
Nov 20 at 21:47
Oh I see. Then it seems the question does not make sense. The question is asking for me to give a condition that makes the subspace <A> unique, where <A> is colspan(A). Since <A> is defined as colspan(A), then by definition <A> is unique?
– Iamanon
Nov 20 at 21:51
@Iamanon yes that's a nonsensical question. If we assume all the columns are linearly independent we at least can say that every element can be written in a unique way as a linear combination of columns. But that's not the same as unicity of the subspace,
– Henno Brandsma
Nov 20 at 21:53
Actually, $<A>$ is defined as the column SPAN (not space) of A. The column space is unique., but I think the column span need not be composed of the span of linearly independent columns. For example, if A is rank 3. Then we can still say that the column span is the span of the 3 columns of A. But this would not be the column space of A. The column space of A would be the span of the two linearly independent columns.
– Iamanon
Nov 20 at 21:55
|
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
The column space is unique as it is defined as the smallest (wrt inclusion) subspace that contains all the columns of $A$ as elements. In this case it also is the set of all linear combinations of columns of $A$, which is minimal (as we can check it is a subspace and any subspace containing the columns must contain their linear combinations).
answered Nov 20 at 21:43
Henno Brandsma
103k346113
Thanks. So if $A$ is full rank, then the column space of A is unique?
– Iamanon
Nov 20 at 21:44
@Iamanon It's always unique, but with full rank it is equal to the maximal subspace (the whole vector space the columns lie in), so $mathbb{R}^n$ in the case of a real-valued $n times m$ matrix.
– Henno Brandsma
Nov 20 at 21:47
Oh I see. Then it seems the question does not make sense. The question is asking for me to give a condition that makes the subspace <A> unique, where <A> is colspan(A). Since <A> is defined as colspan(A), then by definition <A> is unique?
– Iamanon
Nov 20 at 21:51
@Iamanon yes that's a nonsensical question. If we assume all the columns are linearly independent we at least can say that every element can be written in a unique way as a linear combination of columns. But that's not the same as unicity of the subspace,
– Henno Brandsma
Nov 20 at 21:53
Actually, $<A>$ is defined as the column SPAN (not space) of A. The column space is unique., but I think the column span need not be composed of the span of linearly independent columns. For example, if A is rank 3. Then we can still say that the column span is the span of the 3 columns of A. But this would not be the column space of A. The column space of A would be the span of the two linearly independent columns.
– Iamanon
Nov 20 at 21:55
|
show 1 more comment
up vote
1
down vote
accepted
The column space is unique as it is defined as the smallest (wrt inclusion) subspace that contains all the columns of $A$ as elements. In this case it also is the set of all linear combinations of columns of $A$, which is minimal (as we can check it is a subspace and any subspace containing the columns must contain their linear combinations).
answered Nov 20 at 21:43
Henno Brandsma
103k346113
Thanks. So if $A$ is full rank, then the column space of A is unique?
– Iamanon
Nov 20 at 21:44
@Iamanon It's always unique, but with full rank it is equal to the maximal subspace (the whole vector space the columns lie in), so $mathbb{R}^n$ in the case of a real-valued $n times m$ matrix.
– Henno Brandsma
Nov 20 at 21:47
Oh I see. Then it seems the question does not make sense. The question is asking for me to give a condition that makes the subspace <A> unique, where <A> is colspan(A). Since <A> is defined as colspan(A), then by definition <A> is unique?
– Iamanon
Nov 20 at 21:51
@Iamanon yes that's a nonsensical question. If we assume all the columns are linearly independent we at least can say that every element can be written in a unique way as a linear combination of columns. But that's not the same as unicity of the subspace,
– Henno Brandsma
Nov 20 at 21:53
Actually, $<A>$ is defined as the column SPAN (not space) of A. The column space is unique., but I think the column span need not be composed of the span of linearly independent columns. For example, if A is rank 3. Then we can still say that the column span is the span of the 3 columns of A. But this would not be the column space of A. The column space of A would be the span of the two linearly independent columns.
– Iamanon
Nov 20 at 21:55
|
show 1 more comment
up vote
1
down vote
accepted
up vote
1
down vote
accepted
The column space is unique as it is defined as the smallest (wrt inclusion) subspace that contains all the columns of $A$ as elements. In this case it also is the set of all linear combinations of columns of $A$, which is minimal (as we can check it is a subspace and any subspace containing the columns must contain their linear combinations).
answered Nov 20 at 21:43
Henno Brandsma
103k346113
The column space is unique as it is defined as the smallest (wrt inclusion) subspace that contains all the columns of $A$ as elements. In this case it also is the set of all linear combinations of columns of $A$, which is minimal (as we can check it is a subspace and any subspace containing the columns must contain their linear combinations).
answered Nov 20 at 21:43
Henno Brandsma
103k346113
answered Nov 20 at 21:43
Henno Brandsma
103k346113
answered Nov 20 at 21:43
Henno Brandsma
103k346113
answered Nov 20 at 21:43
Henno Brandsma
103k346113
103k346113
Thanks. So if $A$ is full rank, then the column space of A is unique?
– Iamanon
Nov 20 at 21:44
@Iamanon It's always unique, but with full rank it is equal to the maximal subspace (the whole vector space the columns lie in), so $mathbb{R}^n$ in the case of a real-valued $n times m$ matrix.
– Henno Brandsma
Nov 20 at 21:47
Oh I see. Then it seems the question does not make sense. The question is asking for me to give a condition that makes the subspace <A> unique, where <A> is colspan(A). Since <A> is defined as colspan(A), then by definition <A> is unique?
– Iamanon
Nov 20 at 21:51
@Iamanon yes that's a nonsensical question. If we assume all the columns are linearly independent we at least can say that every element can be written in a unique way as a linear combination of columns. But that's not the same as unicity of the subspace,
– Henno Brandsma
Nov 20 at 21:53
Actually, $<A>$ is defined as the column SPAN (not space) of A. The column space is unique., but I think the column span need not be composed of the span of linearly independent columns. For example, if A is rank 3. Then we can still say that the column span is the span of the 3 columns of A. But this would not be the column space of A. The column space of A would be the span of the two linearly independent columns.
– Iamanon
Nov 20 at 21:55
|
show 1 more comment
Thanks. So if $A$ is full rank, then the column space of A is unique?
– Iamanon
Nov 20 at 21:44
@Iamanon It's always unique, but with full rank it is equal to the maximal subspace (the whole vector space the columns lie in), so $mathbb{R}^n$ in the case of a real-valued $n times m$ matrix.
– Henno Brandsma
Nov 20 at 21:47
Oh I see. Then it seems the question does not make sense. The question is asking for me to give a condition that makes the subspace <A> unique, where <A> is colspan(A). Since <A> is defined as colspan(A), then by definition <A> is unique?
– Iamanon
Nov 20 at 21:51
@Iamanon yes that's a nonsensical question. If we assume all the columns are linearly independent we at least can say that every element can be written in a unique way as a linear combination of columns. But that's not the same as unicity of the subspace,
– Henno Brandsma
Nov 20 at 21:53
Actually, $<A>$ is defined as the column SPAN (not space) of A. The column space is unique., but I think the column span need not be composed of the span of linearly independent columns. For example, if A is rank 3. Then we can still say that the column span is the span of the 3 columns of A. But this would not be the column space of A. The column space of A would be the span of the two linearly independent columns.
– Iamanon
Nov 20 at 21:55
Thanks. So if $A$ is full rank, then the column space of A is unique?
– Iamanon
Nov 20 at 21:44
Thanks. So if $A$ is full rank, then the column space of A is unique?
– Iamanon
Nov 20 at 21:44
@Iamanon It's always unique, but with full rank it is equal to the maximal subspace (the whole vector space the columns lie in), so $mathbb{R}^n$ in the case of a real-valued $n times m$ matrix.
– Henno Brandsma
Nov 20 at 21:47
@Iamanon It's always unique, but with full rank it is equal to the maximal subspace (the whole vector space the columns lie in), so $mathbb{R}^n$ in the case of a real-valued $n times m$ matrix.
– Henno Brandsma
Nov 20 at 21:47
Oh I see. Then it seems the question does not make sense. The question is asking for me to give a condition that makes the subspace <A> unique, where <A> is colspan(A). Since <A> is defined as colspan(A), then by definition <A> is unique?
– Iamanon
Nov 20 at 21:51
Oh I see. Then it seems the question does not make sense. The question is asking for me to give a condition that makes the subspace <A> unique, where <A> is colspan(A). Since <A> is defined as colspan(A), then by definition <A> is unique?
– Iamanon
Nov 20 at 21:51
@Iamanon yes that's a nonsensical question. If we assume all the columns are linearly independent we at least can say that every element can be written in a unique way as a linear combination of columns. But that's not the same as unicity of the subspace,
– Henno Brandsma
Nov 20 at 21:53
@Iamanon yes that's a nonsensical question. If we assume all the columns are linearly independent we at least can say that every element can be written in a unique way as a linear combination of columns. But that's not the same as unicity of the subspace,
– Henno Brandsma
Nov 20 at 21:53
Actually, $<A>$ is defined as the column SPAN (not space) of A. The column space is unique., but I think the column span need not be composed of the span of linearly independent columns. For example, if A is rank 3. Then we can still say that the column span is the span of the 3 columns of A. But this would not be the column space of A. The column space of A would be the span of the two linearly independent columns.
– Iamanon
Nov 20 at 21:55
Actually, $<A>$ is defined as the column SPAN (not space) of A. The column space is unique., but I think the column span need not be composed of the span of linearly independent columns. For example, if A is rank 3. Then we can still say that the column span is the span of the 3 columns of A. But this would not be the column space of A. The column space of A would be the span of the two linearly independent columns.
– Iamanon
Nov 20 at 21:55
|
show 1 more comment
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