Question

Prove that if a is a symmetric matrix, then ker a = coker a and img a = coimg a.

a) Symmetric matrices have unique properties that do not relate to kernel and image.

b) The statement is false for all symmetric matrices.

c) For a symmetric matrix, the kernel equals the cokernel, and the image equals the coimage.

d) Symmetric matrices have no relevance to linear algebra concepts.

Answer 1

The statement is false because for symmetric matrices, there's no inherent property that makes the kernel equal to the cokernel, or the image equal to the coimage. The correct answer is option b).

Step-by-step explanation:

The statement to be proven is that for a symmetric matrix a, the kernel (ker a) equals the cokernel (coker a) and the image (img a) equals the coimage (coimg a). However, in the context of infinite-dimensional spaces, such as those studied in functional analysis, the cokernel and coimage could come into play but would not necessarily be equal to the kernel and image.

In finite dimensions, the cokernel of a linear transformation represented by a matrix is the quotient of the codomain by the image of the transformation, which is essentially the dual concept of the kernel but not identical to it.

Thus, it appears that the correct answer to the student's question is b, which states that the given statement is false for all symmetric matrices, with the understanding that the question may involve a misunderstanding of these concepts.