Here is the question I am trying to solve (Matroid Theory, second edition by Martin Oxley, Chapter 1, section 2): Prove that if $C$ is a circuit of a matroid $M$ and $e \in C,$ then $M$ has a basis $B$ such that $C = C(e, B).$ My thoughts: My idea for solving this question is that, given a circui...

I am working on entrance exam problem, wherein I aim to demonstrate that the following sets constitute uncountable sets. (A) The set of all $2\times 2$ real matrices with rational eigenvalues. (B) The set of all real matrices whose row echelon form has rational entries My attempt: Set A is unco...

A forward moving sine wave passes through a uniformly rotating plane orthogonal to it, what will the resulting image on the plane look like?

Here is the question I am trying to solve part (c) in it: $(a)$ Prove that $U_{2,4}$ is representable over a field $\mathbb F$ if anf only if $|\mathbb F| \geq 3.$ $(b)$ For $n \geq 2,$ generalize $(a)$ by finding necessary and sufficient conditions for $U_{2,n}$ to be $\mathbb {F}$-representable...