If n*n matrix, can row space ever be equal to null space?

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Discussion Overview

The discussion centers around the question of whether the row space of an n*n matrix can ever be equal to its null space. Participants explore the concepts of row space and null space, their properties, and implications in linear algebra.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants assert that if a vector belongs to both the row space and null space of a matrix A, it leads to specific implications regarding the vector's properties.
  • One participant notes that the null space and row space of A are complementary subspaces in R^n, suggesting that their dimensions add up to n and that their intersection is trivial.
  • Another participant discusses the case of the zero matrix, stating that its null space spans R^n while its row space is the zero vector, indicating a scenario where the two spaces could be equal.
  • It is mentioned that for any invertible matrix, the row space spans R^n and the null space is the zero vector, further complicating the relationship between the two spaces.
  • One participant concludes that for a singular matrix, the dimensions of the row space and null space must satisfy a specific relationship, leading to the assertion that the only way for the row space and null space to be equal is if both are the zero vector, implying an "empty" matrix scenario.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the row space and null space can be equal. While some agree on the complementary nature of these spaces, others provide specific examples that challenge or refine this understanding, leading to an unresolved discussion.

Contextual Notes

Participants reference specific properties of matrices, such as dimensions and the implications of the zero vector, without reaching a consensus on the broader question of equality between row space and null space.

math_maj0r
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If n*n matrix, can row space ever be equal to null space?

P.S.: this is NOT a homework question. It's a general question to get the concepts straight in my head.
 
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Suppose v is a vector that belongs to the row and null spaces of A, then:

v^T = a^{T}A

And:

Av=\bold 0

Therefore, v^{T}v must be equal to what? And what does this imply regarding v?
 
JSuarez said:
Suppose v is a vector that belongs to the row and null spaces of A, then:

v^T = a^{T}A

And:

Av=\bold 0

Therefore, v^{T}v must be equal to what? And what does this imply regarding v?

What does this symbol mean? a^T
 
The transpose, of course.
 
The nullspace and rowspace of A are complementary subspaces in R^n (i.e. dim of nullspace + dim of rowspace = n and the dot of any vector in the nullspace with any vector in the rowspace will give 0)

vectors in the basis of the nullspace are of the form Ax=0

vectors in the basis of the rowspace are the pivots rows of the row reduced echelon form of A
Consider the zero Matrix
 
Noxide said:
The nullspace and rowspace of A are complementary subspaces in R^n (i.e. dim of nullspace + dim of rowspace = n and the dot of any vector in the nullspace with any vector in the rowspace will give 0)

vectors in the basis of the nullspace are of the form Ax=0

vectors in the basis of the rowspace are the pivots rows of the row reduced echelon form of A



Consider the zero Matrix

Consider the zero Matrix: it's nullspace will span R^n and it's Rowspace will be the zero vector.

Any invertible matrix has a rowspace that spans R^n and it's nullspace will be the zero vector


Any singular matrix will have k vectors in its rowspace basis and z vectors in its nullspace basis such that z + k = n

since the only vector dotted with itself that gives zero is the zero vector then the nullspace and rowspace would have to both be equal to the zero vector to give zero, meaning that the matrix must be "empty" for that to be possible and an "empty" matrix isn't a matrix at all, so there exists no matrix such that nullspace = rowspace
 

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