mathimatikoi.org

Let $W\subset \mathbb{R}^n$ be a linear subspace of dimension at most $n-1$. Which of the following statements are true ??
a)$W$ is nowhere dense
b)$W$ is closed
c)$\mathbb{R}^n-W$ is connected
d)$\mathbb{R}^n-W$ is not connected

No take a sequence $(2+\frac{\sqrt{2}}{n},\sqrt{2})\to (2,\sqrt{2})$

How to find the number of homomorphism of these
a)$$Q_8\to K_4$$
b)$$K_4\to S_n(n\neq 4)$$
c)$$K_4\to S_4$$

For any two symmetric $n\times n$ matrices $A$ and $B$ be their eigenvalues be ordered from largest to smallest . How to prove that for eigenvalues $|\lambda _k^A-\lambda _k^B|\le ||A-B||$ for $1\le k\le n$. Where $\lambda _k^A,\lambda _k^B$ are respective eigenvalues of $A$ and $B$