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, $\partial A$

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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

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