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How they found the left nullspace in each of these examples

  • Thread starter LongApple
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Homework Statement



Part b)

http://www.math.utah.edu/~zwick/Classes/Fall2012_2270/Lectures/Lecture19_with_Examples.pdf
upload_2015-1-22_11-56-16.png



For B
upload_2015-1-22_11-57-0.png


Left nullspace is solution to A ^ T times Y =0
So we have a free variable for the third row so don't we have infinitely many solutions as x3 could be anything?
upload_2015-1-22_11-51-3.png


In this problem's part b) I don't think that they took the transpose of the matrix A.

http://staff.imsa.edu/~fogel/LinAlg/PDF/29 Fundamental Subspaces.pdf
Go the bottom of page 1 under 4) and you'll see "
Perform Gaussian (or better, Gauss-Jordan) elimination on [A | I] to produce [U | B] (or
[R | C]). Claim: the last m – r rows of B (which equal those of C because once we get zeroes
there’s no more work to do in those rows) form ...."

So why isn't the last three rows of 0's in A just the left nullspace?

I am trying to figure out why

Homework Equations




The Attempt at a Solution

 

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  • #2
vela
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Left nullspace is solution to A ^ T times Y =0
So we have a free variable for the third row so don't we have infinitely many solutions as x3 could be anything?
B is a 2x3 matrix, so it maps vectors from ##\mathbb{R}^3## to ##\mathbb{R}^2##. In which of these two vector spaces does the left nullspace reside?

In this problem's part b) I don't think that they took the transpose of the matrix A.

http://staff.imsa.edu/~fogel/LinAlg/PDF/29 Fundamental Subspaces.pdf
Go the bottom of page 1 under 4) and you'll see "
Perform Gaussian (or better, Gauss-Jordan) elimination on [A | I] to produce [U | B] (or
[R | C]). Claim: the last m – r rows of B (which equal those of C because once we get zeroes
there’s no more work to do in those rows) form ...."

So why isn't the last three rows of 0's in A just the left nullspace?
You start with the (untransposed) A and form the augmented matrix [A | I] and perform row operations to produce [U | B]. The basis vectors are the last m-r rows of B. So why are you thinking the rows of A have anything to do with the basis of the left nullspace? It should be clear the last three rows of A can't form basis because they're all 0s.
 
  • #3
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B is a 2x3 matrix, so it maps vectors from ##\mathbb{R}^3## to ##\mathbb{R}^2##. In which of these two vector spaces does the left nullspace reside?


You start with the (untransposed) A and form the augmented matrix [A | I] and perform row operations to produce [U | B]. The basis vectors are the last m-r rows of B. So why are you thinking the rows of A have anything to do with the basis of the left nullspace? It should be clear the last three rows of A can't form basis because they're all 0s.
I don't think so.

But then what does this mean?
https://api.viglink.com/api/click?format=go&jsonp=vglnk_142242965321740&key=6afc78eea2339e9c047ab6748b0d37e7&libId=80df8b60-b64d-4f19-8b03-11259450f0c4&loc=https://www.physicsforums.com/threads/how-they-found-the-left-nullspace-in-each-of-these-examples.793657/#post-4984950&v=1&out=http://staff.imsa.edu/~fogel/LinAlg/PDF/29%20Fundamental%20Subspaces.pdf&ref=https://www.physicsforums.com/&title=How they found the left nullspace in each of these examples&txt=http://staff.imsa.edu/~fogel/LinAlg/PDF/29 Fundamental Subspaces.pdf
Perform Gaussian (or better, Gauss-Jordan) elimination on [A | I] to produce [U | B] (or
[R | C]). Claim: the last m – r rows of B (which equal those of C because once we get zeroes
there’s no more work to do in those rows) form ...."
 
  • #4
vela
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Sorry for the late reply. I've been kinda busy this week.

I simply paraphrased the passage you quoted from the PDF, so I'm not sure what you're disagreeing with. As a result of the row operations, matrix A turns into U, and the identity matrix turns into B. The passage says the last rows of B then form a basis for the left nullspace.
 

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