Why Use the Last Row of the Elimination Matrix for Left Nullspace?

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The discussion centers on the method of finding the basis for the left nullspace of a matrix A by utilizing the last row of the elimination matrix E, as demonstrated by a professor during a class on linear algebra. This approach is not a general property applicable to all matrices but can be effective when the nullspace is one-dimensional, allowing it to be spanned by a single vector. The participants clarify that while this method can simplify the process, it does not universally apply to all row-reduced matrices.

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  • Familiarity with nullspaces and their properties
  • Knowledge of elimination matrices and their role in matrix transformations
  • Concept of left nullspace and its significance in linear systems
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only_huce
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In class my professor row reduced a matrix A into the form U. Then he started to go over how to find the basis for the left nullspace in a matrix A.

Instead of going through the entire process of row reducing the transpose of A and finding its nullspace, he just used the last row of the elimination matrix E as the basis.

Can someone explain why this is so? Is it just a property of the leftnullspace? And can I use it for finding the basis of N(A^T) for any row reduced matrix?
 
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No, that's not a general property. I suspect that this particular nullspace was one-dimensional and so could be spanned by a single vector. That is not generally true.
 

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