📖 Eigenvalue and Eigenvector A scalar $\\lambda$ is an eigenvalue of $T: V \\to V$ if there exists a nonzero vector $v$ such that $T(v) = \\lambda v$. Such $v$ is called an eigenvector. From: Advanced Linear Algebra Learn more: https://advanced-linalg.vercel.app/#/section/10 Explore all courses: https://mathacademy-cyan.vercel.app

💡 Proposition II.10 If a straight line be bisected, and a straight line be added to it in a straight line, the square on the whole with the added straight line and the square on the added straight line both together are double of the square on the half and of the square described on the straight line made up of the half and the added straight line as on one straight line. From: Euclid's Elements Learn more: https://euclid-deploy.vercel.app/#/section/59 Explore all courses: https://mathacademy-cyan.vercel.app

📖 Minimum Distance The minimum distance $d$ of a code is the smallest Hamming distance between distinct codewords. From: intro-discrete Learn more: https://mathacademy-cyan.vercel.app/#/section/20 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Jordan Canonical Form Theorem Every linear operator on a finite-dimensional complex vector space has a unique Jordan canonical form (up to ordering of blocks). Proof: **Existence:** By the primary decomposition theorem, $V = V_1 \\oplus \\cdots \\oplus V_k$ where $V_i$ is the generalized eigenspace for $\\lambda_i$. On each $V_i$, $(T - \\lambda_i I)$ is nilpotent. For nilpotent operators, there exists a Jordan basis giving blocks of the form $J_m(0)$. Combi... From: Advanced Linear Algebra Learn more: https://advanced-linalg.vercel.app/#/section/12 Explore all courses: https://mathacademy-cyan.vercel.app

💡 Proposition I.10 To bisect a given finite straight line. From: Euclid's Elements Learn more: https://euclid-deploy.vercel.app/#/section/10 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Strong Perfect Graph Theorem A graph is perfect if and only if it contains no odd hole and no odd antihole as an induced subgraph. From: Introduction to Graph Theory Learn more: https://west-graphs-deploy.vercel.app/#/section/23 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Ratio Test If $L = \\lim |a_{n+1}/a_n|$, then: $L < 1 \\Rightarrow$ converges absolutely; $L > 1 \\Rightarrow$ diverges; $L = 1 \\Rightarrow$ inconclusive. From: Real Analysis Learn more: https://real-analysis-eta.vercel.app/#/section/7 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$ Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset. Therefore, every element of $A$ is in $B$. Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition. Therefore, every element of $B$ is in $A$. Since $A \\subseteq B$ and $B \\subseteq A... From: Calculus: A Liberal Art Learn more: https://calc-lib-art-deploy.vercel.app/#/section/2 Explore all courses: https://mathacademy-cyan.vercel.app

📖 Holm Step-Down Procedure Sort p-values: $p_{(1)} \\leq \\cdots \\leq p_{(m)}$. Reject $H_{0(j)}$ if $p_{(j)} < \\alpha/(m-j+1)$ for all $j \\leq L$, where $L$ is first index where condition fails. From: Intro to Statistical Learning Learn more: https://mathacademy-cyan.vercel.app/#/section/60 Explore all courses: https://mathacademy-cyan.vercel.app

📖 Definition VII.11 (Prime Number) A prime number is that which is measured by a unit alone. From: Euclid's Elements Learn more: https://euclid-deploy.vercel.app/#/section/179 Explore all courses: https://mathacademy-cyan.vercel.app

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