Question

a) Let a quadratic form have the expression x
2
+ y
2
+ 2z
2
+ 2xy + 3xz with respect to the
standard basis B
1 = f(1; 0; 0); (0; 1; 0); (0; 0; 1)g. Find its expression with respect to the
basis B
2 = f(1; 1; 1); (0; 1; 0); (0; 1; 1)g (3)
b) Consider the quadratic form
Q : 2x
2
4xy + y
2
+ 4xz + 3z
2
i) Find a symmetric matrix A such that Q = X
t
AX .
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. (5)