Tooltip Test Page - Mathnotes

This page tests the tooltip functionality for math block references.

Basic Definitions

Definition A vector space over a field $F$ is a set $V$ together with two operations: - Vector addition: $V \times V \to V$ - Scalar multiplication: $F \times V \to V$ satisfying the following axioms: 1. $(V, +)$ is an abelian group 2. Scalar multiplication is associative: $a(bv) = (ab)v$ 3. Distributive laws hold 4. Identity: $1v = v$ for all $v \in V$

Theorem: Dimension Theorem Let $V$ be a finite-dimensional vector space and $W$ be a subspace of $V$. Then: $$\dim(W) \leq \dim(V)$$ with equality if and only if $W = V$. :::proof Let $\{w_1, \ldots, w_k\}$ be a basis for $W$. Since $W \subseteq V$, these vectors are linearly independent in $V$. By the basis extension theorem, we can extend this to a basis of $V$.

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Here are various types of references to test tooltips:

Let's also test some cross-file references if they exist:

Proposition A subset $W$ of a vector space $V$ is a subspace if and only if: 1. $0 \in W$ 2. $W$ is closed under addition 3. $W$ is closed under scalar multiplication :::example The set of all $n \times n$ symmetric matrices forms a subspace of $M_n(\mathbb{R})$.

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These should show error styling: - @nonexistent-label - @type:wrong-label

Definition The derivative of a function $f$ at a point $a$ is defined as: $$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ provided this limit exists.