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







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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







    share|cite|improve this question
























      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







      share|cite|improve this question













      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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 20 at 21:34









      Iamanon

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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).






          share|cite|improve this answer





















          • 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
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          active

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          up vote
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          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).






          share|cite|improve this answer





















          • 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















          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).






          share|cite|improve this answer





















          • 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













          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).






          share|cite|improve this answer












          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).







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 20 at 21:43









          Henno Brandsma

          103k345111




          103k345111












          • 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










          • @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


















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