Question

In: Advanced Math

Consider the following quadratic forms q(x1, x2) = 3x1^2 − 6x1x2 + 11x2^2 and r(x1, x2,...

Consider the following quadratic forms

q(x1, x2) = 3x1^2 − 6x1x2 + 11x2^2 and

r(x1, x2, x3) = x1^2 − x2^2+x3^2+ 2x1x2 − 6x1x3+2x2x3,

on R 2 and R 3 , respectively. In both cases do the following.

(a) Find the symmetric matrix A representing the quadratic form.

(b) Find a corresponding orthogonal matrix P of eigenvectors of that matrix.

(c) Write down the maximum and minimum values of the quadratic form over the unit vectors (in R 2 and R 3 , respectively).

Please include steps and answer a), b) and c)

Expert Solution

Colby Messinger answered 2 years ago

Determine if the following set forms a subspace in R^2. The set is (x1,x2)^t ,in other...

Determine if the following set forms a subspace in R^2. The set is (x1,x2)^t ,in other words the column vector [x1,x2]. Can you go through each axiom and show your work?, I have a lot of difficulty with these types of questions and I want to make sure I understand.Thank you in advance.

Is the following map linear? a) F(x1,x2,x3)=(0,0) b) L:R2→R2 defined by L(x1,x2)=(3x1−2x2,x2) c) f:R→R defined by...

Is the following map linear? a) F(x1,x2,x3)=(0,0) b) L:R2→R2 defined by L(x1,x2)=(3x1−2x2,x2) c) f:R→R defined by f(x)=2x

Consider the following three consumption bundles (X1,X2)=(10,10) ; (X1,X2)=(15,10) ; (X1,X2)=(3000,8).

Answer each of the following statements True/False/Uncertain. Give a full explanation of your answer including graphs where appropriate. (When in doubt, always include a fully labeled graph.)A) Consider the following three consumption bundles (X1,X2)=(10,10) ; (X1,X2)=(15,10) ; (X1,X2)=(3000,8). Non-satiation implies that (15,10) is preferred to (10,10) but does not imply that (3000,8) is preferred to (10,10).B) It is not theoretically possible for two indifference curves to cross if the preference relations they are based on satisfy the assumptions of completeness,...

A consumer’s utility function is U(x1,x2)=3x1+x21/3. If the consumer weakly prefers the bundle (x1’,x2’) to the...

A consumer’s utility function is U(x1,x2)=3x1+x21/3. If the consumer weakly prefers the bundle (x1’,x2’) to the bundle (x1’’,x2’’), will he necessarily also weakly prefer the bundle (x1’+1,x2’) to the bundle (x1’’+1,x2’’)?

1/Laura's preferences over commodities x1 and x2 can be represented by U(x1,x2)=min{3x1, x2}. She maximizes her...

1/Laura's preferences over commodities x1 and x2 can be represented by U(x1,x2)=min{3x1, x2}. She maximizes her utility subject to her budget constraint. Suppose there is an increase in p1. There is an income effect but not a substitution effect of this price change. There are both income and substitution effects of this price change. There is a substitution effect but not an income effect of this price change. It is unclear whether the consumer will buy more or less x1...

Consider the linear system of equations below 3x1 − x2 + x3 = 1 3x1 +...

Consider the linear system of equations below 3x1 − x2 + x3 = 1 3x1 + 6x2 + 2x3 = 0 3x1 + 3x2 + 7x3 = 4 i. Use the Gauss-Jacobi iterative technique with x (0) = 0 to find approximate solution to the system above up to the third step ii. Use the Gauss-Seidel iterative technique with x (0) = 0 to find approximate solution to the third step

Consider the following linear programming problem Maximize $1 X1 + $2 X2 Subject To 2 X1...

Consider the following linear programming problem Maximize $1 X1 + $2 X2 Subject To 2 X1 + X2 ≤ 8 Constraint A X1 + X2 ≤ 5 Constraint B X1, X2 ≥ 0 Constraint C Note: Report two digits after the decimal point. Do NOT use thousands-separators (,) 1 - Which of the following is the correct standard maximization form for the above linear programming problem AnswerCorrectNot Correct AnswerCorrectNot Correct AnswerCorrectNot Correct AnswerCorrectNot Correct Z -X1 - 2 X2 =...

Solve the following problems using the two phase method: max 3x1 + x2 S.t. x1 −...

Solve the following problems using the two phase method: max 3x1 + x2 S.t. x1 − x2 ≤ −1 −x1 − x2 ≤ −3 2x1 + x2 ≤ 4 x1, x2 ≥ 0 Please show all steps

Amy has utility function u (x1, x2) = min { 2(x1)^2x2, x1*(x2)^2 }. Derive Amy's demand...

Amy has utility function u (x1, x2) = min { 2*(x1)^2*x2, x1*(x2)^2 }. Derive Amy's demand function for x1 and x2. For what values (if any) of m, p1, and p2 are the goods gross complements or gross substitutes of each other?

x1 + x2 - 2x4 = 2 x1 + x2 + 2x3 + 6x4 + x5...

x1 + x2 - 2x4 = 2 x1 + x2 + 2x3 + 6x4 + x5 = 8 −2x1 - 2x2 + x3 + 8x4 = −1 3x3 + 12x4 + 2x5 = 9 Let the linear system be given. a. Find the reduced row eelon form of the combined matrix (augmented matrix) of the system. b. Is the system consistent? If the system is consistent, find the overall solution of the system. c. Do all the solutions of the...