The Mathematics

The Implicit Function Theorem For Functions from Rn to Rn Examples 1 – The Math

Recall from The Implicit Function Theorem for Functions from Rn to Rn page that if $A subseteq mathbb{R}^{n+k}$ is open and $mathbf{f} : A to mathbb{R}^n$ is a continuously differentiable function on $A$ ($mathbf{f}$ is $C^1$ on $A$) then if there exists an $(mathbf{x}_0; mathbf{t}_0) in A$ for which $mathbf{f}(mathbf{x}_0; mathbf{t}_0) = 0$ and the …

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The Inclusion Chart for Special Types of Integral Domains – The Math

Below is an inclusion chart showing the various inclusions of the algebraic structures of: commutative rings, integral domains, unique factorization domains, principal ideal domains, Euclidean domains, and Fields: (1) begin{align} quad mathrm{Commutative : Rings} supset mathrm{Integral : Domains} supset mathrm{UFDs} supset mathrm{PIDs} supset mathrm{EDs} supset mathrm{Fields} end{align} The math online

Injectivity and Surjectivity of the Adjoint of a Linear Map

In the following two propositions we will see the connection between a linear map $T$ being injective/surjective and the corresponding adjoint matrix $T^*$ being surjective/injective. Proposition 1: Let $V$ and $W$ be finite-dimensional nonzero inner product spaces and let $T in mathcal L (V, W)$. Then $T$ is injective if and only if $T^*$ is …

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