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Let S: M_n(R) -> M_n(R) defined by: S(A) = A - A^T

Now, I need to do three things for this question. a) Show S is a linear transformation b) Find Ker(S) and describe it c)) Find Rng(s) and describe it

For part a, i just showed that S(A)=A-A^T satisfies addition and scalar multiplication, which wasn't too difficult of a task. For addition, i made S(A+B) = (A+B) - (A+B)^T then worked out the algebra till i got S(A+B) on the right side of the equation. For scalar multiplication id di the same thing with S(cA) = cA-cA^T.

I am not sure how to start the second 2 sections though (b&c). I know how to get the kernel (nullspace) of a matrix, but not for a question like this. Can anyone help me?

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For the second one think of the symmetric matrix in $M_n(R)$ – Babak S. Mar 1 '13 at 16:01
how does that help arrive at a specific kernel? i feel like this question is not giving me enough information – Johnathon Svenkat Mar 1 '13 at 16:15
up vote 1 down vote accepted

For the kernel, note that you have $$A-A^\mathrm{T} =0 \iff A=A^\mathrm{T}$$ Therefore a matrix is in the kernel if and only if it is symmetric. What is the dimension of the space of $n\times n$ symmetric matrices?

For the range, note that the image of the mapping is always skew-symmetric. Next note that $$\mathrm{M}_n(\mathbb{R})=\mathrm{Sym}_n(\mathbb{R})\oplus\mathrm{Skew}_n(\mathbb{R})$$ Use this together with rank-nullity to show that the mapping is surjective onto $\mathrm{Skew}_n(\mathbb{R})$.

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