💡 Proposition VII.8 If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole. From: Euclid's Elements Learn more:

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The foundational text of Western mathematics. Study all 13 books of Euclid

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📐 Lagrange If $H \\leq G$ and $G$ is finite, then $|H|$ divides $|G|$. The index is $[G:H] = |G|/|H|$. Proof: Cosets partition $G$ into $[G:H]$ sets, each of size $|H|$. Thus $|G| = [G:H] \\cdot |H|$. From: intro-discrete Learn more:

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📐 Diagonalizability via Minimal Polynomial $T$ is diagonalizable if and only if its minimal polynomial is a product of distinct linear factors. Proof: **(⇒)** If $T$ is diagonalizable with eigenvalues $\\lambda_1, \\ldots, \\lambda_k$, then $(T - \\lambda_1 I) \\cdots (T - \\lambda_k I) = 0$ on each eigenbasis vector, hence on all of $V$. The minimal polynomial divides $(x - \\lambda_1) \\cdots (x - \\lambda_k)$. **(⇐)** If $m(x) = (x - \\lamb... From: Advanced Linear Algebra Learn more:

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A rigorous treatment of linear algebra based on Hoffman & Kunze, covering vector spaces, linear transformations, and canonical forms.

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