A kernel, also known as null space, is the set of all vectors that when multiplied by a matrix result in the zero vector. It is important in matrix calculations because it represents the solutions to homogeneous linear equations, and can help determine linear independence and dimensionality of a matrix.
Yes, a matrix can have multiple kernels if it has more than one set of linearly independent vectors that span the null space.
To find a matrix with a given kernel, you can set up a system of equations using the given vectors as the basis for the kernel. Then, solve for the remaining variables in the matrix to create a matrix that satisfies the given conditions.
The kernel and the range of a matrix are complementary subspaces. This means that the dimension of the kernel plus the dimension of the range equals the dimension of the original vector space.
Yes, a matrix can have a kernel of dimension 0 if its columns are all linearly independent, meaning that the only solution to the equation Ax=0 is the zero vector. In this case, the matrix is invertible and its null space is only the zero vector.