The discussion centers on the relationship between the kernel of a matrix \( A \) and the kernel of the expression \( A^2 + A \). It is established that \( \text{ker}(A^2 + A) \) is a subset of \( \text{ker}(A) \), as demonstrated by breaking down the equations \( A^2x = 0 \) and \( Ax = 0 \). The participants emphasize the importance of formalizing the argument and suggest starting from \( x \in \text{ker}(A) \) for clarity. Additionally, the factorization \( A^2 + A = A(A + I) = (A + I)A \) is noted as a useful approach in understanding the relationship.
PREREQUISITESStudents and educators in linear algebra, mathematicians focusing on matrix theory, and anyone seeking to deepen their understanding of kernel relationships in linear transformations.
Candice said:Homework Statement
- What is the relation between the kernel of A and the kernel of (A^2 + A)?
Homework Equations
The Attempt at a Solution
Break into A^2x = 0 and Ax = 0. We know Ax = 0 because that's the kernel of A, ker(A^2x) is subset of ker(A) so ker(A^2 + A) is a subset of ker (A)?