hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
('T polnt) Solve the equation AX(D + BX)-1 = C for X. Assume that all matrices are n x n and invertible as needed. You can enter the inverse of a matrix A as A^(-1). X =
Problem 25 please -Sesin(2x)-9ecos(2x). 21. W = Span(B), where Br(x2e-4x , xe®, e-4x); f(x)--5x2r" + 2e-4-1e 22. W= Span(B),where B= ({x25, x5*, 5x)); f(x)--4x2 5x+9s5x-2(5x). 3 W Span(B), where B (Exsin(2x), xcos(2x), sin(2x), cos(2x)y): f(x) = 4x sin(2x) + 9x cos(20-5 sin(2x) + 8 cos(2x). 24, In Exercise 21 of Section 3.6, we constructed the matrix [D, of the derivative operator D on W- Span(B), where B e sin(bx), e" cos(bx)): Dls a a. Find [D 1g and [D'lg: Observe...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), by minimizing lle subject to Ar-b. (a) Show that if A Rmxn is full (row) rank, where m n, then AA is invertible. Then show that r.-A7(AAT)-ibis a solution to Ax = b. (b) Along with part (a) and the solution aAT(AA)-b, show that l thus, z is the optimal solution to the minimization problem. and...
Can a quadratic equation aX^2 + bX + c = 0, where a, b, and c are rationals, has one rational solution and one irrational solution? Prove or disprove with justification.
Problem 5. A subset A c Rn is an affine subspace of Rn if there exists a vector b є R', and a underlying vector subspace W of Rn such that (a) Describe all the affine subspaces of R2 which are not vector subspaces of R2. (b) Consider A є Rnxn, b є Rn and the system of linear equations Ax-b Prove that (i) if Ar= b is consistent, then its solution set is an affine subspace of Rn with...
The equation for a parabola has the form y=ax^2+bx+c, where a, b, and c are constants and a≠0. Find an equation for the parabola that passes through the points (−1,0), (−2,−1), and (−6,15). Answer: y =
w(t, x) = f(bx + ct) for a transverse wave, where b and c are constants and f( ) is some arbitrary function. Is this a traveling wave model? If so, in what direction does it travel? +x -x The wave does not move In the y or z direction because the wave is transverse
algebra 2 Solving the quadratic equation using the quadratic formula ax²+bx+c 01 x= -htb² - 4ac Ex3 - 2x² +3 x =4-15 X X - 2x²-x =-15 +15 +15 - 2x2-x+15= 0 A = -2 D- C= 15
Consider the utility function u(x) = ax + b e^cx where a, b, c are positive scalars. (a) Compute the coefficient of absolute risk aversion. (b) Describe the risk attitude represented by u(x) and how it changes as x increases. (c) Write down the equations to determine the certainty equivalent and the risk premium of a gamble X for an individual with initial wealth w > 0. (d) What is the sign of the risk premium? How does the risk...