I have a theoretical question. Why are non-square matrices not invertible? I am running into a lot of doubts like this in my introductory study of linear algebra.

Let un =Hn − log n u n = H n log n. If we prove that the sequence (un)n (u n) n is convergent (it's limit is the Euler constant γ γ) then we conclude that Hn = Θ(log n) H n = Θ (log n).

Aug 15, 2015 · This may be a stupid question but I just want clarification about the use of the name of this rule. Well, most of the time what I see is L'Hospital's Rule, like in Baby Rudin and …

How can I convert ax2 + bx + c = 0 a x 2 + b x + c = 0 to a FOIL-style (x + d)(x − e) = 0 (x + d) (x e) = 0 equation? I have an equation in a computer program that I'm currently solving with the …

A Lie group is simultaneously a differentiable manifold. As I understand it, the Lie group is generated via exponentiation of the generators of the Lie algebra. It is intuitively clear to me …

Sep 27, 2020 · X is compact set iff X closed and pre-compact. X is called pre-compact if $\\overline{X}$ is compact. But in some texts, I found that X pre-compact is totally bounded. Is …

How can I show that U(n) U (n) is a subgroup of SO(2n) S O (2 n)? I can see how we can identify Cn C n with R2n R 2 n by mapping a + ib ↦ (a, b) a + i b ↦ (a, b), but after that I'm a bit …

Oct 15, 2020 · Consider a unitary group U(n) = {A ∈ GLn(C): A∗A = I} U (n) = {A ∈ G L n (C): A ∗ A = I}. I want to show that U(n) U (n) is a compact group. First, I can observe that U(n) ⊆Mn(C) U …

Since O(n) O (n) is a subgroup of U(n) U (n), we can consider the quotient space Ln = U(n)/O(n) L n = U (n) / O (n). The quotient space is homotopic to the (n n -th) Lagrangian Grassmannian, …

I want to show that U(n) U (n) is a regular submanifold of GL(n,C) G L (n, C), for this I figured I use the regular level set theorem. For U(n) U (n) is the preimage of the identity I I under the …