Attempt I have come across this sequence of questions whilst revising for my linear algebra exam,I have tried to answer these questions (with my attempt attached) but I am in no way sure that I have got it right. Any help would be much appreciated, thanks!
(a) Let be the vector space of 3×3 real symmetric matrices (i.e. matrices which satisfy transpose =). Show that ⟨,⟩:→ℝ given by ⟨,⟩=Tr() is an inner product on
(b) Consider the following matrices in : =[(1,0,1),(0,1,0),(1,0,0)] and =[(1,0,1),(0,−3,0),(1,0,0)] Find a matrix that is orthogonal to both and with respect to the inner product.
To show that $<,>$ is a inner porduct,
You just have to show that $<,>$ is a bilinear symetric definite positive form.
For the second question you don't have to compute the eigen values of A or B.
You must find $C \in V$ such that $Tr(AC)=Tr(BC)=0$
You have to write what it means with $A,B,C$ coefficient, then you will find some conditions on the coefficients of $C$ in order to have $C$ orthogonal to $A$ and $B$