# How they found the left nullspace in each of these examples

## Homework Statement

Part b)

http://www.math.utah.edu/~zwick/Classes/Fall2012_2270/Lectures/Lecture19_with_Examples.pdf For B 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? 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

## Answers and Replies

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.

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?f...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 ...."

vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
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.