Question

. Show that ax + by + cz = 0 a1x + b1y + c1z = 0 always has solutions other than x = y = z = 0. (Hint: use rank)

Answer #1

First, create 3 equations of the form ax+by+cz=d , where a, b,
c, and d are constants (integers between – 5 and 5). For example, x
+ 2y – 2= -1 . Perform row operations on your system to obtain a
row-echelon form and the solution. Go to the 3D calculator website
GeoGebra: www.geogebra.org/3d?lang=pt and enter each of the
equations. After you have completed this first task, choose one of
the following to complete your discussion post.
1. Reflect on...

Suppose y^2 = x^3+ax+b with a, b ∈ Q defines an elliptic curve.
Show that there is another equation Y^2 = X^3 + AX + B with A, B ∈
Z whose solutions are in bijection with the solutions to y^2 =
x^3+ax+b.

Exercise 2.4 Assume that a system Ax = b of linear equations has
at least two distinct solutions y and z.
a. Show that xk = y+k(y−z) is a solution for every
k.
b. Show that xk = xm implies k = m. [Hint:
See Example 2.1.7.]
c. Deduce that Ax = b has infinitely many solutions.

Prove: If f(x) = anx^n + an−1x^n−1 + ··· + a1x + a0 has integer
coefficients with an ? 0 ? a0 and there are relatively prime
integers p, q ∈ Z with f ? p ? = 0, then p | a0 and q | an . [Hint:
Clear denominators.]

Suppose that the incircle of triangle ABC touches AB at Z, BC at
X, and AC at Y . Show that AX, BY , and CZ are concurrent.

Show that the beta of a portfolio is the weighted average of the
beta's of the portfolio's assets. Hint:
Cov(aX+Y,Z)=aCox(X,Z)+Cov(Y,Z)

Let A be an nxn matrix. Show that if Rank(A) = n, then Ax = b
has a unique solution for any nx1 matrix b.

For the given function u(x, y) = cos(ax) sinh(3y),(a >
0);
(a) Find the value of a such that u(x, y) is harmonic.
(b) Find the harmonic conjugate of u(x, y) as v(x, y).
(c) Find the analytic function f(z) = u(x, y) + iv(x, y) in
terms of z.
(d) Find f ′′( π 4 − i) =?

A random variable X has a probability function f(x) = Ax, 0 ≤ x
≤ 1, 0, otherwise.
a. What is the value of A? (Hint: intigral -inf to inf f(x)dx=
1.)
b. Compute P(0less than x less than 1/3)
c. Compute the cdf. of X.
d. Compute E(X).
e. Compute V(X).

Find two linearly independent solutions of
2x2y′′−xy′+(−2x+1)y=0,x>0 of the form
y1=xr1(1+a1x+a2x2+a3x3+⋯) y2=xr2(1+b1x+b2x2+b3x3+⋯) where
r1>r2.
Enter r1= a1= a2= a3= r2= b1= b2= b3=
Note: You can earn partial credit on this problem.

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