Recent content by slamminsammya

Homework Statement Find all points for which the curves x^2+y^2+z^2=3 and x^3+y^3+z^3=3 share the same tangent line. Homework Equations Sharing the same tangent line amounts to having the same derivative. The constraint then is that 3x^2+3y^2+3z^2=2x+2y+2z. The points must obviously also...

The following question was posed on an old qualifying exam for linear algebra: Suppose A is an n by n complex matrix, and that A has spectral radius <1 (the eigenvalue with largest norm has norm <1). Show that A^n approaches 0 as n goes to infinity. The solution is easy when the eigenspace...

I completely agree with your method here, but I don't think that x^2-1 is the characteristic polynomial of A. The characteristic polynomial is the determinant of A-cI, and for any n by n matrix the characteristic polynomial must therefore be of degree n. Forgive me if I am misunderstanding, but...

Out of curiosity, it seems like there should be a way of proving this without invoking the spectral theorem for symmetric matrices. Does anyone know a proof?

Yes it does... Thanks

Yes, I have been able to show that 1 is the only eigenvalue, but I just can't seem to get from there to showing that A therefore fixes each vector. The proof that 1 is the only eigenvalue goes like this: Suppose Ax=cx for a scalar c. Then A(Ax)=c^2x=x, so that the only possible eigenvalues are...

Homework Statement Suppose that A is a real n by n matrix which is orthogonal, symmetric, and positive definite. Prove that A is the identity matrix.Homework Equations Orthogonality means A^t=A^{-1}, symmetry means A^t=A, and positive definiteness means x^tAx>0 whenever x is a nonzero...

I am in a similar situation. Does anyone have any recommendations for a good refresher book in multivariable calculus? I.e. one that will cover the important concepts but in a manner that is perhaps more sophisticated than what you encounter in a first year undergrad course?

Homework Statement Let K be a field, and f,g are relatively prime in K[x]. Show that f-yg is irreducible in K(y)[x]. Homework Equations There exist polynomials a,b\in K[x] such that af+bg=u where u\in K. We also have the Euclidean algorithm for polynomials. The Attempt at a...

Given K a field, and f\in K[x] an irreducible (monic) polynomial. Does it follow that the field K[x]/\left<f\right> is an algebraic extension of K?

Suppose that f:\mathbb{R} \to \mathbb{R} is continuous and f'(x_0) exists for some x_0 . Does it follow that f' exists for all x such that |x-x_0|<r for some r>0 ?

Thanks!

Homework Statement The question is from Koblitz's "P-adic Numbers, p-adic Analysis, and Zeta Functions". It asks me to prove that ord_p(p^N!)= \sum _{i=1}^{N-1}p^i. Homework Equations Firstly it would be important to define ord_px. This is defined to be the exponent of the highest power...