Homework Statement
Let \lambda_0 \in \mathbb{C} be an eingenvalue of the n \times n matrix A with algebraic multiplicity m , that is, is an m-nth zero of \det{A-\lambda I} . Consider the perturbed matrix A+ \epsilon B , where |\epsilon | \ll 1 and B is any n \times n matrix...
Homework Statement
Prove \sin{10} , in degrees, is irrational.
Homework Equations
None, got the problem as is.
The Attempt at a Solution
Im kinda lost.
Homework Statement
I've been reading some introductory PDE books and they always seem to motivate the search for solutions of the first order quasilinear PDE by the method of characteristics, by introducing a flow model; i was thinking, could someone give an example of a phenomenon modeled by...
Sorry for the inactivity, my computer decided to self-destruct under the heat.
Well, that (a_1,a_2,...,a_n) does not lie in our rectangle centered about the point x is not much of a problem, since said rectangle was constructed inside our original and arbitrary rectangle, not necessarily...
Let R be an open rectangle such that x \in R , R=(a_1,b_1)\times ... \times (a_n,b_n) . If x=(x_1,...,x_n) , we construct an open rectangle R' with sides smaller than 2\min{(b_i - x_i, x_i-a_i)} for 1\leq i \leq n , and centered about the point x . By construction R' \subset R and...
Homework Statement
I have been self studying Spivak's Calculus on Manifolds, and in chapter 1, section 2 (Subsets of Euclidean Space) there's a problem in which you have to find the interior, exterior and boundary points of the set
U=\{x\in R^n : |x|\leq 1\}.
While it is evident that...
We do not know if f and g are continuous, we only assume them to be integrable, so it is not necessarily true that
\int (f-\lambda g)^{2}=0
implies (f-\lambda g)^{2}=0, since f-\lambda g could be zero except at an isolated number of points (it's integral would still be zero but the...
Homework Statement
In an effort to keep me from spending all summer lying on the couch, I recently started reading Michael Spivak's Calculus on Manifolds; while working on problem 1-6 I got stuck on a technical detail and I was wondering if anyone could provide a little insight.
Problem 1-6...
As other people pointed out, do the dot product for all three vectors and you'll get a system of 4 unknowns and 3 equations whose answer is most likely a 4-3=1-dimensional subspace of R^4. The answer should be a line.
I have been doing some self study on differential equations using Tom Apostol's Calculus Vol. 1. and I got stuck on a problem (problem 12, section 8.5, vol. 1).
Homework Statement
Let K be a non zero constant. Suppose P and Q are continuous in an open interval I.
Let a\in I and b a real...
Homework Statement
Let A be a square matrix with right inverse B. To prove A has a left inverse C and that B = C.
Homework Equations
Matrix multiplication is asociative (AB)C=A(BC).
A has a right inverse B such that AB = I
The Attempt at a Solution
I dont really know where to...
Homework Statement
Let f:A\subset{\mathbb{R}}^{n}\mapsto \mathbb{R} be a linear function continuous a \vec{0} . To prove that f is continuous everywhere.
Homework Equations
If f is continuous at zero, then \forall \epsilon>0 \exists\delta>0 such that if \|\vec{x}\|<\delta then...