I think I got it.
A = PDP^(-1)
A^T = (P^-1)^T * D^T * P^T
D^T = D
(P^-1)^T = (P^T)^-1
A^T = (P^T)^-1 * D P^T
Since P is invertible, it has linearly independent columns and so does P^T.
So let P^T = M
A^T = MDM^-1
Therefore A^T is diagonalizable.
Therefor A^T has n linearly independent...
I still can't piece the last bit together. I realize that A^T has linearly independent columns, and rows that aren't scalar multiples of each other. I also see that the Transpose has the same diagonal entries as the original. However, the systems I'm setting up with transpose to find...
I mean technically you're in a correct form of infinity over infinity, but thinking broadly: I learned l'hopistal's rule in single variable calculus. You're writing g(x,y) as g(x) by mistake (at least I think so). I don't know if you need to use partials in three space as such, but I always...
Homework Statement
Show that if an nxn matrix A has n linearly independent eigenvectors, then so does A^T
The Attempt at a Solution
Well, I understand the following:
(1) A is diagonalizable.
(2) A = PDP^-1, where P has columns of the independent eigenvectors
(3) A is...
Homework Statement
Solve the reducible 2ODE. Assume x, y and/or y' positive where helpful.
y^3 * y'' = 1
The Attempt at a Solution
Well, I tried what I normally would do for x being missing.
p = (dy/dx); y'' = p'p = (dp/dy)(dy/dx)
So
y^3 p'p = 1
p(dp/dy) = y^(-3)...
So I have the integral from t[0, 6pi] of
<0,0,185-(9t/6pi)> dot <-20sint, 20cost, 15/pi>dt
= 2775/pi - (45 x pi x t) / 2
Which gives me 4,100 foot lb.
Is the difference in the work done really that large? 166500 - 4100?
Homework Statement
Part 1: A 160lb man carries a 25lb paint can up a spiral staircase, which has radius 20 feet, completes 3 revolutions, and has final height 90 feet. What is the work done?
Part 2: This time, the man's paint can leaks at a constant rate such that he loses 9lbs of paint...