(1) If A is a real symmetric matrix,show that tr(AA)≧0,and tr(AA)>0 if A≠O.Thus the trace defines a positive definite scalar product on the space of real symetric matirces.
(2) Let V be the vector space of real n×n symmetric matrices.
What is dimV? What is the dimension of the subspace W consisting of those matrices A such that tr(A)=0? What is the dimension of the orthogonal complement W^⊥ relative to the positive definite scalar product of part (1)?

の(2)の最後についての質問です。
dimV=(n^2-n)/2+n=(n^2+n)/2.
dimW={v∈V;a_11x_1+a_22x_2+…+a_nnx_n=0}=dimA-rank(a_11 a_22 … a_nn)=(n^2+n)/2-1.
となり,最後は
dimW^⊥はdimW^⊥=dimV-dimW=(n^2+n)/2-((n^2+n)/2-1)=1
としてみたのですが「relative to the positive definite scalar product of part (1)」
の意味わかりません。
dimW^⊥=1は間違いなのでしょうか?