Recent content by unawareness

Homework Statement Let A be the set that contains all rational numbers, but not zero. Let (a,b),(c,d) \in A×A. Let (a,b)\tilde{}(c,d) if and only if ad = bc. Prove that \tilde{} is an equivalence relation on A×A.Homework Equations The Attempt at a Solution The solution just needs to show...

Yeah, I have no clue where this diagonal matrix stuff is going. I'm probably doing everything wrong. I wound up with $$ P = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} $$ I think P is singular, so I won't be able to find a P-1.

Then set it equal to zero to get λ=0 and λ=1/2 but there's two 1/2's, so it's still a valid diagonalizable matrix? Also, the theories behind diagonalizing a matrix are completely unknown to me. I don't know what it means and I don't know why doing it is supposed to be a good thing.

I've seen and done it, but I can't recall the method. I'm looking it up, but it's only vaguely familiar still. If A is a 3x3 matrix with 3 different eigenvalues, then it is diagonalizable? And to diagonalize it... you take 3 eigen vectors and combine them into a matrix P. Then P-1AP =...

S isn't quite ℝ3. The uhh... 3rd dimension is always equal to one. It's like an x-y plane shifted up one unit along the 3rd dimension. I don't know what you call that. But that method would work then? If I took two random points with the 3rd dimension equal to 1, then that would tell me if...

I've been playing with it some more. Any arbitrary vector seems to converge to the point (-2,2,1) when iterated under the transformation. And on the last part where I chose x = 2 and y = 5, I mean x is the vector (2,0,0) and y is the vector (0,5,0). I'm not sure if I'm allowed to choose these...

Homework Statement Show that the following linear transformation matrix is a contraction mapping. \begin{bmatrix} 0.5 & 0 & -1 \\ 0 & 0.5 & 1 \\ 0 & 0 & 1 \end{bmatrix} I don't know how to make that into a matrix, but it is a 3x3 matrix. The first row is [.5 0 -1] the second row is [0...