Linear Algebra: Nullity of Matrix Product Question

Click For Summary
SUMMARY

The discussion centers on proving that for any m*n matrix A and any n*n matrix B, the nullity of matrix A is less than or equal to the nullity of the product AB. Participants clarify that nullity is defined as the dimension of the kernel of a matrix, and it is expressed as nullity A = n - rank A. The conversation emphasizes the relationship between nullity and rank, asserting that nullity AB cannot exceed nullity A due to the properties of linear transformations.

PREREQUISITES
  • Understanding of matrix dimensions (m*n and n*n matrices)
  • Familiarity with the concepts of nullity and rank in linear algebra
  • Knowledge of linear transformations and their properties
  • Basic proficiency in mathematical proofs and logical reasoning
NEXT STEPS
  • Study the relationship between nullity and rank in linear algebra
  • Explore the implications of the Rank-Nullity Theorem
  • Investigate properties of matrix multiplication and its effects on nullity
  • Learn about linear transformations and their kernel in depth
USEFUL FOR

Students of linear algebra, mathematicians, and educators looking to deepen their understanding of matrix theory and its applications in various fields.

workerant
Messages
40
Reaction score
0
Prove that for any m*n matrix A and any n*n matrix B, nullity A is less than or equal to nullity AB.

Sorry Dick: I know nullity of A is n-rank A and this is always greater than or equal to 0 and I figured nullity AB is like nullity A * nullity B, which are both greater than or equal to 0. Is this in the right direction?
 
Last edited:
Physics news on Phys.org
Any thoughts on this? Even if you have no idea how to solve it you are supposed to at least thrash around and show an attempt. Such are the forum rules.
 

Similar threads

Finding properties of a linear transformation
  • · Replies 2 ·
Replies
2
Views
2K
I don't understand why the rank = n - Rank-nullity theorem - nullity
  • · Replies 2 ·
Replies
2
Views
2K
Finding Kernel, Image, Rank, Nullity of Matrix
  • · Replies 1 ·
Replies
1
Views
2K
Proving Nullity in Linear Algebra: A < AB for m*n and n*n Matrices
  • · Replies 1 ·
Replies
1
Views
3K
Rank of A & Construction of W: Linear Algebra Homework
  • · Replies 4 ·
Replies
4
Views
2K
Is the Kernel of a 9x10 Matrix with Rank 5 Four-Dimensional?
  • · Replies 1 ·
Replies
1
Views
2K
Linear maps (rank and nullity)
  • · Replies 4 ·
Replies
4
Views
2K
Full Rank Matrix: Determinant Condition | Rank-Nullity Theorem
  • · Replies 4 ·
Replies
4
Views
2K
Proof involving Rank Nullity Theorem
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad Confused about small detail in rank-nullity theorem
  • · Replies 1 ·
Replies
1
Views
3K