Mathematics

2011 midterm exam in Algebraic Structures main 1. If \(S = \left\{ \begin{bmatrix} a & 0 & b \\ 0 & c & d \\ 0 & 0 & a \end{bmatrix} \;\middle|\; a,b,c,d \in \mathbb{R} \right\}\), show that S is a subring of \(M_3(\mathbb{R})\), a 3 by 3 matrix over \(\mathbb{R}\). Solution: Clearly, \(S \subseteq…

2011 midterm exam in Topology main 1. Suppose \(A \subseteq B\) and B is finite, show that A is also finite. Solution: B is finite implies that there is a natural number \(n\) such that there exists a one-to-one correspondence between the points of B and the numbers {1, 2, 3, …, n}. This further…