Rank of AB: How nxn Matrices A & B Determine Rank

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SUMMARY

The discussion centers on the relationship between the ranks of two nxn matrices A and B, specifically stating that rank(AB) is greater than or equal to rank(A) + rank(B) - n. This is grounded in the Rank-Nullity Theorem, which defines the rank of linear transformations. The rank of matrix B is the dimension of its image in vector space V, while the rank of matrix A is the dimension of its image in vector space W. The conclusion emphasizes the necessity for elements in AB(U) to be derived from both A and B's transformations.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically matrix rank
  • Familiarity with the Rank-Nullity Theorem
  • Knowledge of linear transformations and their properties
  • Basic understanding of vector spaces and dimensions
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  • Study the Rank-Nullity Theorem in detail
  • Explore applications of matrix rank in linear transformations
  • Learn about the implications of rank in systems of linear equations
  • Investigate the properties of nxn matrices in relation to their ranks
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Mathematicians, students of linear algebra, and anyone involved in theoretical computer science or engineering who seeks to understand the implications of matrix rank in linear transformations.

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check that, for any nxn matrices A,B then rank(AB) (> or =) rank A +rank(B)-n
 
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Are you familiar with the Rank-Nullity Theorem?
 
Suppose B:U-> V and A:V->W. The rank of B is the dimension of B(U) in V, the rank of A is the dimension of A(V)->W, and the rank of AB is the dimension of AB(U) in W. In order that z be in AB(U), it must be in A(V) so that z= A(y) for some y in U. And y must be in B(U) so that y= Bx for some x in U.
 

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