# Orthogonal Complement

1. Nov 2, 2008

### ahamdiheme

1. The problem statement, all variables and given/known data

Consider the vector space $$\Re$$nxn over $$\Re$$, let S denote the subspace of symmetric matrices, and R denote the subspace of skew-symmetric matrices. For matrices X,Y$$\in$$$$\Re$$nxn define their inner product by <X,Y>=Tr(XTY). Show that, with respect to this inner product,
R=S$$\bot$$

2. Relevant equations

Definition of inner product
Definition of orthogonal compliment
Definition of symmetric matrix
Definition of skew symmetric matrix

3. The attempt at a solution
If i can show that
R-S$$\bot$$=0
will it be sufficient and how do i go about it?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 2, 2008

### HallsofIvy

Staff Emeritus
What do you mean by $R- S^{\bot}= 0$? To show that $R= S^{\bot}$ you must show that the inner product of any member of R with any member of S is 0, that's all.

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