Question #46094

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)

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)

Expert's answer

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