Question

a) Let a quadratic form have the expression x^2+y^2+2z^2+2xy+3xz with respect to the standard basis B1 ={(1,0,0),(0,1,0),(0,0,1)}.
Find its expression with respect to the basis B2 ={(1,1,1),(0,1,0),(0,1,1)}


b) Consider the quadratic form Q:2x^2−4xy+y^2+4xz+3z^2
i) Find a symmetric matrix A such that Q = XtAX.
ii) Find the orthogonal canonical reduction of the quadratic form.
iii) Find the principal axes of the form.
iv) Find the rank and signature of the form