calculus review 01
basic
linear transformations
measure of matrices: \(\left \rVert A\right \lVert_F^2\)
triagle inq in matrices: \(\rVert AB \lVert\le \rVert A\lVert\rVert B\lVert\)
neighborhood of \(x\), exist an open ball init
closure \(\bar{A}\), smallest close set that contains A
interior \(\mathring{A}\), largest open set that is contained in A
boundary of subset, \(\partial A\)
convergence of sequence, in terms of coordinates
limits of multivariable functions: continuity is preserved under dot product operation
continuity: the preimage of a neighborhood of \(f(x)\) is also a neighborhood of x
uniform continuity: linear transformations are uniform continuity
convergence of the sum of series (vectors): absolute(norm in vector cases) convergence implies convergence
complext exponentials
- complex exponential series converges: \(e^z=1+z+\frac{z^2}{2!}+\dots=\sum_{k=0}^\infty \frac{z^k}{k!}\), since the absolute series converges
euler formular: \(e^{it}=cost+isint\)
geometric series of matrices:
\(S=I+A+A^2+\dots\) converges to \((1-A)^{-1}\) if \(\lVert A \rVert \lt 1\)
the set of invertable n by n matrices is open
bounded: subset \(X\subset \R^n\) is bounded if it is contained in some ball centered at the origin
compact: nonempty subset \(C\subset \R^n\) is compact if it is closed and bounded
important theorem
Bolzano-Weierstrass theorem: a compact set C contains a seq, then that seq has a convergent sub seq whose limit is in C