📐 Fundamental Theorem of Linear Maps If $V$ is finite-dimensional and $T \\in \\mathcal{L}(V, W)$, then $\\operatorname{range} T$ is finite-dimensional and $\\dim V = \\dim \\operatorname{null} T + \\dim \\operatorname{range} T$. Proof: Let $u_1, \\ldots, u_m$ be a basis of $\\operatorname{null} T$. Extend to a basis $u_1, \\ldots, u_m, v_1, \\ldots, v_n$ of $V$. Then $Tv_1, \\ldots, Tv_n$ is a basis of $\\operatorname{range} T$, giving $\\dim V = m + n = \\dim \\operatorname{null} T + \\dim \\operatorname{range} T$. From: linalg-axler Learn more:

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📖 Splitting Field Let $K$ be a field and let $f(x) = 0$ be a polynomial with coefficients in $K$. A splitting field of $f$ over $K$ is a field $K(a, b, c, \\ldots)$ containing $K$ and a complete set of roots $a, b, c, \\ldots$ of $f(x) = 0$. By the Corollary, the splitting field equals $K(t)$ for a single Galois resolvent $t$. From: gal-edwards Learn more:

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💡 The Significance of Aura Colors Each color in the aura corresponds to a specific quality of soul life. Blue tones indicate devotion and religious feeling; red expresses passion and anger; yellow radiates from intellectual activity; green relates to adaptability and empathy; violet indicates spiritual aspiration. The clarity, brightness, and stability of the colors reveal the degree to which these qualities are purified and de... From: steiner-GA90a Learn more:

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💡 Proposition 6.1.7 (Uniqueness of Limits) Let $(a_n)$ be a real sequence. If $(a_n)$ converges to $L$ and also to $L\ Proof: Suppose $(a_n)$ converges to both $L$ and $L' \\neq L$. Let $\\varepsilon := |L - L'|/3 > 0$. There exists $N$ with $|a_n - L| \\leq \\varepsilon$ for $n \\geq N$, and $M$ with $|a_n - L'| \\leq \\varepsilon$ for $n \\geq M$. For $n := \\max(N, M)$: $|L - L'| \\leq |L - a_n| + |a_n - L'| \\leq... From: tao-analysis-1 Learn more:

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📖 Definition: Pivot A pivot is the first nonzero entry in a row of a matrix in echelon form. The pivot positions determine the structure of the solution space. From: Linear Algebra Learn more:

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📐 Simple Algebraic Extensions Let $K$ be a given field and let $G(X)$ be an irreducible polynomial with coefficients in $K$. Then one can construct a field $K(t)$ such that: (1) $K(t)$ contains $K$, (2) $K(t)$ contains an element $t$ with $G(t) = 0$, and (3) every element of $K(t)$ can be expressed as a polynomial $b_0 + b_1 t + \\cdots + b_\\nu t^\\nu$ where $\\nu < \\deg G$. Moreover, any two such fields are naturally iso... Proof: Let $R$ be the set of all polynomials in $X$ with coefficients in $K$. Two elements are congruent mod $G$ if their difference is divisible by $G(X)$. The quotient $L$ is a ring. The mapping $k \\mapsto$ [class of constant $k$] embeds $K$ into $L$. The class of $X$ is a root of $G$ in $L$. The Euc... From: gal-edwards Learn more:

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💡 Proposition I.18 In any triangle the greater side subtends the greater angle. 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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💡 Proposition III.23 On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side. From: Euclid's Elements Learn more:

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📐 Lines in Homogeneous Coordinates A line in $\\mathbb{RP}^2$ is the set of points $[x : y : z]$ satisfying $ax + by + cz = 0$ for some $(a, b, c) \\neq (0, 0, 0)$. The line can be represented by $[a : b : c]$. From: Four Pillars of Geometry Learn more:

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Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📖 Carmichael Number A Carmichael number is composite $n$ that is a Fermat pseudoprime to all bases $a$ with $\\gcd(a,n) = 1$. From: Algebraic Number Theory Learn more:

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📐 Successive Derivatives of a Polynomial For a polynomial of degree $n$, the $(n+1)$th derivative and all subsequent derivatives are zero. From: Beginner Calculus Learn more:

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Calculus Made Easy - An interactive introduction to calculus

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📐 Inverse Function Theorem If f: R^n → R^n is C¹ and Df(a) is invertible, then f is a local diffeomorphism near a. From: analysis-pugh Learn more:

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💡 The Threefold Soul Between the ego and the three lower bodies lie three soul members: the Sentient Soul (the ego working in the astral body), the Intellectual Soul or Mind Soul (the ego working in the etheric body), and the Consciousness Soul (the ego working in the physical body). These three soul members represent stages of the ego\ From: steiner-GA90a Learn more:

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📐 Spectral Theorem for Symmetric Matrices A real symmetric matrix $A$ has all real eigenvalues, and eigenvectors from different eigenspaces are orthogonal. $A$ can be diagonalized as $A = Q\\Lambda Q^T$ with orthogonal $Q$. Proof: Real eigenvalues: For $A\\mathbf{x} = \\lambda\\mathbf{x}$ with $A = A^T$, $\\bar{\\mathbf{x}}^TA\\mathbf{x} = \\lambda\\|\\mathbf{x}\\|^2$ and $\\bar{\\mathbf{x}}^TA^T\\mathbf{x} = \\bar{\\mathbf{x}}^TA\\mathbf{x} = \\bar{\\lambda}\\|\\mathbf{x}\\|^2$. Thus $\\lambda = \\bar{\\lambda}$, so $\... From: Linear Algebra Learn more:

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📐 Archimedean Property For every real number x, there exists a natural number n such that n > x. Equivalently, for any ε > 0, there exists n ∈ N with 1/n < ε. From: analysis-pugh Learn more:

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📐 When is 2 a QR? $(2/p) = 1$ if $p \\equiv \\pm 1 \\pmod{8}$; $(2/p) = -1$ if $p \\equiv \\pm 3 \\pmod{8}$. From: Algebraic Number Theory Learn more:

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📖 OP_CHECKLOCKTIMEVERIFY (BIP-65) OP_CLTV compares the top stack item to the transaction nLockTime. If nLockTime < stack value, the script fails. Enables absolute time locks: "cannot spend until block N or time T". From: bips Learn more:

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💡 Conservative Design I wanted to design a system that could do everything that might be needed in the future, without having to go back and change the fundamentals. From: satoshi Learn more:

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💡 Fermat Prime Condition If $2^n + 1$ is prime, then $n$ must be a power of $2$. Proof: Suppose $n$ has an odd divisor $k > 1$, so $n = km$ for some $m$. Using the factorization for odd $k$: $x^k + 1 = (x+1)(x^{k-1} - x^{k-2} + \\cdots + 1)$: $2^n + 1 = 2^{km} + 1 = (2^m)^k + 1 = (2^m + 1)(\\cdots)$. Since $2^m + 1 > 1$, this shows $2^n + 1$ is composite. Therefore $n$ can h... From: numbers-geometry Learn more:

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📐 Upper-Triangular Matrix Every operator on a finite-dimensional nonzero complex vector space has an upper-triangular matrix with respect to some basis. From: linalg-axler Learn more:

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