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)?
The kernel of a matrix A is the set of all vectors that when multiplied by A result in the zero vector. In other words, it is the solution space to the homogeneous equation Ax=0.
The kernel of A is closely related to the original matrix A as it represents the set of vectors that are mapped to the zero vector by A. In other words, the kernel is a subset of the domain of A.
The kernel of A provides information about the nullity of A, which is the dimension of the kernel. This tells us how many linearly independent vectors are mapped to the zero vector by A, and can provide insight into the invertibility and rank of A.
To find the kernel of A, we can solve the homogeneous equation Ax=0 using techniques such as Gaussian elimination or matrix row operations. The resulting solutions will form the basis for the kernel.
Yes, it is possible for the kernel of A to be empty. This would occur when there are no vectors that are mapped to the zero vector by A, meaning the nullity of A is zero. In this case, the matrix A would be one-to-one and have a full rank.