Dimensions of the Four Subspaces

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February 12th 2011, 07:58 PM
alexmahone

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Dimensions of the Four Subspaces

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Could someone please explain the answers for Column space and Nullspace of $A^{T}$ ?

Thanks.
February 12th 2011, 09:21 PM
dwsmith

Quote:

Originally Posted by alexmahone

http://www.mathhelpforum.com/math-he...ar-algebra.jpg

Could someone please explain the answers for Column space and Nullspace of $A^{T}$ ?

Thanks.

The column space will be the row space of A (I think).

The null space of the transpose is the left null space of A.
February 12th 2011, 09:26 PM
alexmahone

Quote:

Originally Posted by dwsmith

The column space will be the row space of A (I think).

The dimension of the column space is equal to that of the row space. But the column space is not equal to the row space in this problem.

Quote:

Originally Posted by dwsmith

The null space of the transpose is the left null space of A.

I know that. But I don't know how the textbook has found Nullspace of $A^{T}$ in this particular problem.
February 12th 2011, 09:27 PM
dwsmith

Quote:

Originally Posted by alexmahone

I know that. But I don't know how the textbook has found Nullspace of $A^{T}$ in this particular problem.

$\mathbf{x}^T\mathbf{A}=\mathbf{0}^T$
February 12th 2011, 09:31 PM
dwsmith

Are you sure the column space of A^T isn't the basis of the row space of A? I would think it is that.

It is stated on http://www.ltcconline.net/greenl/cou...ctors/rank.htm that that the column space of A^T is the row space of A.

I don't see why it wouldn't be the same the other way around.
February 13th 2011, 04:23 AM
HallsofIvy

I cannot see the "A" that apparently was posted so I don't know if we are talking about matrices or general linear transformations or what dimensions. It is true that the "row space" of a linear transformation A is the same as (strictly speaking "is isomorphic to") the "column space" for finite dimensions. For infinite dimensions (such as function spaces) two spaces can have the same dimension but not be isomorphic.
February 13th 2011, 07:27 AM
alexmahone

Quote:

Originally Posted by HallsofIvy

I cannot see the "A" that apparently was posted so I don't know if we are talking about matrices or general linear transformations or what dimensions. It is true that the "row space" of a linear transformation A is the same as (strictly speaking "is isomorphic to") the "column space" for finite dimensions. For infinite dimensions (such as function spaces) two spaces can have the same dimension but not be isomorphic.

I reposted the image.

A has been factorised as $LU=E^{-1}R$ . The author apparently calculates the Column space of A from $E^{-1}$ and the Nullspace of $A^{T}$ from E. I don't understand how he does that.
February 13th 2011, 09:22 AM
LoblawsLawBlog

I think he's using rank-nullity and then just making sure he gets the correct number of independent vectors. He knows the column space of $A$ is two, and the first and third columns of $A$ are the ones he lists and they're independent. I'm not sure if this is obvious from $E^{-1}$ , but I suspect it might be because of the nice factorization. I forget all the details about the relationships between these bases and how they change when you do elementary operations.

As for the nullspace of $A^T$ , it looks like he just interprets matrix multiplication as acting on the rows of $A$ and then finds a vector in the nullspace of $A^T$ . It's in this nullspace because he considers left multiplication on $A$ . Since this nullspace has dimension 1, there's a basis.

Is this Strang? I know he talks about this in one of his MIT lectures online. That should answer everything.
February 13th 2011, 09:22 AM
dwsmith

Quote:

Originally Posted by alexmahone

I reposted the image.

A has been factorised as $LU=E^{-1}R$ . The author apparently calculates the Column space of A from $E^{-1}$ and the Nullspace of $A^{T}$ from E. I don't understand how he does that.

The author tells you exactly what I told you in post 4 except the author uses y instead of x.

The left null space of A are the vectors that satisfies $\mathbf{x}^T\mathbf{A}=\mathbf{0}^T$ which is the null space of A^T