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📐 Unique Representation If $\\{v_1, \\ldots, v_n\\}$ is a basis for $V$, then every vector $v \\in V$ can be written uniquely as $v = c_1 v_1 + \\cdots + c_n v_n$. Proof: **Existence:** Since the basis spans $V$, every $v$ is a linear combination. **Uniqueness:** Suppose $v = \\sum c_i v_i = \\sum d_i v_i$. Then: $$\\sum (c_i - d_i) v_i = \\mathbf{0}$$ By linear independence, $c_i - d_i = 0$ for all $i$, so $c_i = d_i$. From: Advanced Linear Algebra Learn more:

Advanced Linear Algebra | Magic Internet Math

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

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📐 Differentiability Implies Continuity If $f$ is differentiable at $c$, then $f$ is continuous at $c$. From: Real Analysis Learn more:

Real Analysis | Math Academy

A rigorous introduction to real analysis covering limits, continuity, differentiation, and integration through formal mathematical proofs.

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Jordan Block A Jordan block $J_k(\\lambda)$ is a $k \\times k$ matrix with $\\lambda$ on the diagonal, $1$s on the superdiagonal, and $0$s elsewhere. From: Advanced Linear Algebra Learn more:

Advanced Linear Algebra | Magic Internet Math

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

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Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Two Squares Theorem A positive integer $n$ is a sum of two squares if and only if every prime factor $p \\equiv 3 \\pmod{4}$ appears to an even power in the factorization of $n$. From: Disquisitiones Arithmeticae Learn more:

Disquisitiones Arithmeticae | Math Academy

Gauss

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Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.