Quick Linear Algebra-True or false type

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SUMMARY

The discussion centers on the relationship between the nullity of matrices A and B and their product AB. Specifically, it questions whether nullity(AB) equals nullity(A) multiplied by nullity(B) for an m*n matrix A and an n*n matrix B. Participants suggest testing this with specific values, such as m=n=3, to explore the validity of the statement. The consensus leans towards the idea that nullity(A) multiplied by nullity(B) does not necessarily equal nullity(AB), particularly in cases where both nullity(A) and nullity(B) are greater than 1.

PREREQUISITES
  • Understanding of matrix dimensions and types (m*n and n*n matrices)
  • Familiarity with the concept of nullity in linear algebra
  • Basic knowledge of matrix multiplication
  • Experience with testing mathematical statements using specific examples
NEXT STEPS
  • Research the properties of nullity in linear algebra
  • Learn about the Rank-Nullity Theorem and its implications
  • Explore examples of matrix multiplication and their nullities
  • Investigate counterexamples where nullity(AB) does not equal nullity(A) * nullity(B)
USEFUL FOR

Students studying linear algebra, educators teaching matrix theory, and mathematicians exploring properties of matrix operations.

workerant
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Quick Linear Algebra---True or false type

Homework Statement



If A is a m*n matrix and B is a n*n matrix, does nullity AB= nullity A* nullity B?

I think that it does, but I am not sure. I don't need a proof of this or anything, this is just a fact I need to know in order to try and prove something more complicated.
 
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Are you sure null(A)*null(B) is always small enough to be equal to null(AB)? Try, e.g. m=n=3 and see if you can get null(A)=null(B)=2

If you're stuck, check the spoiler

For an even more trivial case, m=n=2, A=B=0 the zero matrix.
 

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