Can the Null Space of a Matrix Be the Same as Its Transpose?

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SUMMARY

The null space of a matrix cannot be the same as that of its transpose if the matrix is not square. The null space, defined as the set of vectors x such that Ax = 0, is only applicable when matrix A is square. In cases where A is symmetric, the null space of A and its transpose AT are identical, as the matrix equals its transpose. Therefore, understanding the relationship between a matrix and its transpose is crucial for determining their null spaces.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically null space and kernel.
  • Familiarity with matrix operations, including multiplication and transposition.
  • Knowledge of symmetric matrices and their properties.
  • Basic comprehension of vector spaces and their dimensions.
NEXT STEPS
  • Study the properties of symmetric matrices and their implications on null spaces.
  • Learn about the Rank-Nullity Theorem and its applications in linear algebra.
  • Explore examples of non-square matrices and their null spaces.
  • Investigate the definitions and properties of vector spaces in greater detail.
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Students of linear algebra, mathematicians, and anyone interested in understanding the relationships between matrices and their transposes, particularly in the context of null spaces.

complexhuman
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hmmm...I have problems understanding this...how can the null space if a matrix(not necessarily a square) be the same as that of its transpose?

Thanks in advance
 
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If the matrix is not square, then this is impossible. The null space of a matrix A consists of vectors x such that Ax = 0. If A is not square, and Ax is defined (i.e. you are allowed to multiply A and x) then ATx is not even defined. I'm not sure what you're asking though. In general, the null space of a matrix is not the same if it as the null space of its transpose. However, certainly if the matrix is symmetric then its kernel is the same as the kernel of its transpose, since the matrix is its own transpose.
 

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