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

Learn more about our help with Assignments: Engineering

## Comments

## Leave a comment