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

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In summary, the conversation discusses the topic of finding the basis for the left nullspace in a matrix A. The professor demonstrated how to use the last row of the elimination matrix E as the basis instead of going through the entire process of finding the nullspace of the transpose of A. However, this is not a general property and may only apply to cases where the nullspace is one-dimensional. It cannot be used for finding the basis of N(A^T) for all row reduced matrices.
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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.
 

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

What is the left nullspace?

The left nullspace of a matrix A is the set of all vectors x such that Ax = 0. In other words, it is the set of all solutions to the equation Ax = 0.

How is the left nullspace related to the row space and column space?

The left nullspace is orthogonal to both the row space and the column space of a matrix A. This means that the left nullspace is perpendicular to all the rows and columns of A, and any vector in the left nullspace will produce a dot product of 0 with any vector in the row or column space.

What is the dimension of the left nullspace?

The dimension of the left nullspace is equal to the number of linearly independent columns in A. This is because the number of linearly independent columns is equal to the number of free variables in Ax=0, and each free variable corresponds to a basis vector in the left nullspace.

When is the left nullspace empty?

The left nullspace is empty if and only if the matrix A has full column rank, meaning that all the columns are linearly independent. In this case, the only solution to Ax=0 is the trivial solution x=0, and therefore the left nullspace is empty.

Can the left nullspace contain non-zero vectors?

Yes, the left nullspace can contain non-zero vectors. In fact, any non-zero vector in the left nullspace is orthogonal to all the rows and columns of A, and can be used to find solutions to the equation Ax=0. These non-zero vectors in the left nullspace are also known as null vectors or null space vectors.

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