(a)  Define the quadratic form qA(x1, · · · , xn) associated to a real symmetric n × n matrix A.

(a)  Define the quadratic form qA(x1, · · · , xn) associated to a real symmetric n × n
matrix A. 

(b)  Show that if two real symmetric matrices A, B are congruent then there is a
linear change of variables to x′1,··· ,x′n such that qA(x1,··· ,xn) = qB(x′1,··· ,x′n). 

(c)  Show that the following quadratic form on R3 is not positive definite:
q(x,y,z)=x^2 +4xz+3y^2 +4yz+z^2. 

(d)  Let V be a real vector space with basis v1,v2 and define a dot product by v1 ·v1 =1, v1 ·v2 =v2 ·v1 =λ, v2 ·v2 =2,

where λ ∈ R is fixed. For what values of λ does (V, ·) become an inner product space with the stated dot products? 

5. (Hint: you may wish to diagonalise the associated quadratic form q(x,y)=x^2 +2xλy+2y^2.)