Physlib.Mathematics.LinearMaps

Let $V$ be a vector space (or module) over the rational numbers $\mathbb{Q}$. A homogeneous quadratic equation is a map $f : V \to \mathbb{Q}$ such that for any scalar $a \in \mathbb{Q}$ and any vector $v \in V$, the function satisfies the homogeneity property $f(a \cdot v) = a^2 f(v)$.

This instance allows a homogeneous quadratic equation $f \in \text{HomogeneousQuadratic } V$ to be treated as a function from a vector space $V$ (over $\mathbb{Q}$) to the rational numbers $\mathbb{Q}$. It provides the mechanism to evaluate the equation at a vector $v \in V$ to produce the value $f(v) \in \mathbb{Q}$.

Let $V$ be a vector space (or module) over the rational numbers $\mathbb{Q}$. For any homogeneous quadratic map $f: V \to \mathbb{Q}$, any scalar $a \in \mathbb{Q}$, and any vector $S \in V$, the identity $f(a \cdot S) = a^2 f(S)$ holds.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$. This instance allows a symmetric bilinear form $f \in \text{BiLinearSymm } V$ to be treated as a function from $V$ to the space of $\mathbb{Q}$-linear maps from $V$ to $\mathbb{Q}$ (the dual space). Specifically, for any vector $v \in V$, the evaluation $f(v)$ is a linear functional such that for any $w \in V$, $f(v)(w)$ returns a value in $\mathbb{Q}$.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$. Given a function $f : V \times V \to \mathbb{Q}$, the definition `BiLinearSymm.mk₂` constructs a symmetric bilinear map on $V$ if $f$ satisfies linearity in its first argument and symmetry. Specifically, it requires: 1. Scalar multiplication in the first factor: $f(a \cdot S, T) = a \cdot f(S, T)$ for all $a \in \mathbb{Q}$ and $S, T \in V$. 2. Addition in the first factor: $f(S_1 + S_2, T) = f(S_1, T) + f(S_2, T)$ for all $S_1, S_2, T \in V$. 3. Symmetry (swap): $f(S, T) = f(T, S)$ for all $S, T \in V$. The linearity in the second factor is automatically derived from these properties.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$. For any symmetric bilinear form $f$ on $V$, any scalar $a \in \mathbb{Q}$, and any vectors $S, T \in V$, it holds that $f(a \cdot S, T) = a \cdot f(S, T)$.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$. For any symmetric bilinear form $f$ on $V$ and for any vectors $S, T \in V$, the identity $f(S, T) = f(T, S)$ holds.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$. For any symmetric bilinear form $f$ on $V$, any scalar $a \in \mathbb{Q}$, and any vectors $S, T \in V$, the identity $f(S, a \cdot T) = a \cdot f(S, T)$ holds.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$. For any symmetric bilinear form $f$ on $V$ and any vectors $S_1, S_2, T \in V$, the identity $f(S_1 + S_2, T) = f(S_1, T) + f(S_2, T)$ holds.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$. For any symmetric bilinear form $f$ on $V$ and any vectors $S, T_1, T_2 \in V$, the identity $f(S, T_1 + T_2) = f(S, T_1) + f(S, T_2)$ holds.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$. For a symmetric bilinear form $f$ on $V$ and a fixed vector $T \in V$, this definition represents the linear functional $V \to \mathbb{Q}$ mapping $S \mapsto f(S, T)$.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$. For any symmetric bilinear form $f$ on $V$ and any vectors $S, T \in V$, the value of the form $f(S, T)$ is equal to the evaluation of the linear functional $f.\text{toLinear}_1(T)$ at the vector $S$, where $f.\text{toLinear}_1(T)$ denotes the linear map $V \to \mathbb{Q}$ derived by fixing the second argument of $f$ as $T$.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ and let $f$ be a symmetric bilinear form on $V$. For any natural number $n$, any vector $T \in V$, and any family of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$, the following equality holds: $$f\left(\sum_{i=0}^{n-1} S_i, T\right) = \sum_{i=0}^{n-1} f(S_i, T)$$ where the left-hand side is the bilinear form applied to the sum of the vectors $S_i$ and the vector $T$, and the right-hand side is the sum of the values of the form applied to each individual $S_i$ and $T$.

Let $V$ be a vector space (typically over $\mathbb{Q}$ in this context) and let $f$ be a symmetric bilinear form on $V$. For any natural number $n$, any vector $T \in V$, and any family of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$, the following equality holds: $$f(T, \sum_{i=0}^{n-1} S_i) = \sum_{i=0}^{n-1} f(T, S_i)$$ where the left-hand side is the bilinear form applied to $T$ and the sum of the vectors $S_i$, and the right-hand side is the sum of the results of applying the form to $T$ and each individual $S_i$.

Let $V$ be a vector space over the field of rational numbers $\mathbb{Q}$. Given a symmetric bilinear form $\tau$ on $V$, this definition constructs a homogeneous quadratic map $f: V \to \mathbb{Q}$ defined by $f(v) = \tau(v, v)$ for every vector $v \in V$.

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ and let $\tau$ be a symmetric bilinear form on $V$. Let $Q: V \to \mathbb{Q}$ be the associated homogeneous quadratic map defined by $Q(v) = \tau(v, v)$. For any vectors $S, T \in V$, the following identity holds: $$Q(S + T) = Q(S) + Q(T) + 2\tau(S, T)$$

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. A homogeneous cubic equation (or map) is a function $f: V \to \mathbb{Q}$ such that for any scalar $a \in \mathbb{Q}$ and any element $v \in V$, the condition $f(a \cdot v) = a^3 f(v)$ is satisfied.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. A homogeneous cubic map $f$ (an element of the type `HomogeneousCubic V`) can be treated as a function $f : V \to \mathbb{Q}$. This allows an instance of a cubic equation to be applied directly to an element $v \in V$ to obtain a value in $\mathbb{Q}$.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any homogeneous cubic map $f: V \to \mathbb{Q}$, any scalar $a \in \mathbb{Q}$, and any element $S \in V$, the identity $f(a \cdot S) = a^3 f(S)$ holds.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. An element $f$ of the type `TriLinearSymm V`, representing symmetric trilinear forms on $V$, can be treated as a function $f : V \to (V \to_{\mathbb{Q}} V \to_{\mathbb{Q}} \mathbb{Q})$. This allows a symmetric trilinear form to be applied sequentially to elements of $V$ to obtain a linear map, and eventually a scalar in $\mathbb{Q}$, reflecting the curried representation of a trilinear map.

Let $V$ be a vector space (or module) over the field of rational numbers $\mathbb{Q}$. Given a function $f : V \times V \times V \to \mathbb{Q}$, the constructor `TriLinearSymm.mk₃` defines a symmetric trilinear form on $V$ provided that $f$ satisfies linearity in its first argument and two specific swap symmetries. Specifically, it requires: 1. **Linearity in the first factor**: * Scalar multiplication: $f(a \cdot S, T, L) = a \cdot f(S, T, L)$ for all $a \in \mathbb{Q}$ and $S, T, L \in V$. * Addition: $f(S_1 + S_2, T, L) = f(S_1, T, L) + f(S_2, T, L)$ for all $S_1, S_2, T, L \in V$. 2. **Symmetry between the first and second factors**: $f(S, T, L) = f(T, S, L)$ for all $S, T, L \in V$. 3. **Symmetry between the second and third factors**: $f(S, T, L) = f(S, L, T)$ for all $S, T, L \in V$. Because the map is symmetric, linearity in the second and third arguments is automatically satisfied by the linearity in the first.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L \in V$, the result of the form is invariant under the swapping of the first and second arguments, such that $f(S, T, L) = f(T, S, L)$.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L \in V$, the result of the form is invariant under the swapping of the second and third arguments, such that $f(S, T, L) = f(S, L, T)$.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L \in V$, the form is invariant under the swapping of the first and third arguments, such that $f(S, T, L) = f(L, T, S)$.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$, any scalar $a \in \mathbb{Q}$, and any vectors $S, T, L \in V$, the form satisfies the scalar homogeneity property in its first argument, such that $f(a \cdot S, T, L) = a f(S, T, L)$.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$, any scalar $a \in \mathbb{Q}$, and any vectors $S, T, L \in V$, the form satisfies the scalar homogeneity property in its second argument: $$f(S, a \cdot T, L) = a f(S, T, L).$$

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$, any scalar $a \in \mathbb{Q}$, and any vectors $S, T, L \in V$, the form satisfies the scalar homogeneity property in its third argument, such that: $$f(S, T, a \cdot L) = a f(S, T, L).$$

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$ and any vectors $S_1, S_2, T, L \in V$, the form is additive in its first argument, such that $$f(S_1 + S_2, T, L) = f(S_1, T, L) + f(S_2, T, L).$$

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$ and any vectors $S, T_1, T_2, L \in V$, the form is additive in its second argument, such that $$f(S, T_1 + T_2, L) = f(S, T_1, L) + f(S, T_2, L).$$

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L_1, L_2 \in V$, the form is additive in its third argument, such that $$f(S, T, L_1 + L_2) = f(S, T, L_1) + f(S, T, L_2).$$

Let $V$ be a vector space over the field of rational numbers $\mathbb{Q}$. Given a symmetric trilinear form $f$ on $V$ and two fixed vectors $T, L \in V$, the definition $f.\text{toLinear}_1(T, L)$ is the linear map $V \to \mathbb{Q}$ that sends a vector $S \in V$ to the scalar value $f(S, T, L)$. This is obtained by fixing the second and third arguments of the trilinear form.

Let $V$ be a vector space over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L \in V$, the value $f(S, T, L)$ is equal to the value of the linear map $f.\text{toLinear}_1(T, L)$ applied to $S$. Here, $f.\text{toLinear}_1(T, L)$ represents the linear map $V \to \mathbb{Q}$ obtained by fixing $T$ and $L$ as the second and third arguments of the trilinear form.

Let $V$ be a vector space (or module) over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$, any fixed vectors $T, L \in V$, and any finite collection of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$, the form satisfies linearity in its first argument: $$f\left(\sum_{i} S_i, T, L\right) = \sum_{i} f(S_i, T, L)$$ Here, the application $f(S, T, L)$ represents the evaluation of the trilinear form on the three vectors.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$, any fixed vectors $T, L \in V$, and any finite collection of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$, the form satisfies linearity in its second argument: $$f\left(T, \sum_{i} S_i, L\right) = \sum_{i} f(T, S_i, L)$$ Here, the application $f(T, S, L)$ represents the evaluation of the trilinear form on the three vectors.

Let $V$ be a module over the rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$, and for any vectors $T, L \in V$ and a finite collection of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$, the form satisfies linearity in its third argument: $$f(T, L, \sum_{i} S_i) = \sum_{i} f(T, L, S_i)$$ Here, the curried application $f(T, L, S)$ represents the evaluation of the trilinear form on the three vectors.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. For any symmetric trilinear form $f$ on $V$ and any finite collections of vectors $\{S_i\}_{i < n_1}$, $\{T_k\}_{k < n_2}$, and $\{L_l\}_{l < n_3}$ in $V$, the form satisfies multilinearity across all three arguments simultaneously: $$f\left(\sum_{i} S_i, \sum_{k} T_k, \sum_{l} L_l\right) = \sum_{i} \sum_{k} \sum_{l} f(S_i, T_k, L_l)$$ where the application $f(S, T, L)$ represents the evaluation of the trilinear form on the three vectors.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. Given a symmetric trilinear form $\tau$ on $V$, this definition constructs a homogeneous cubic map $f: V \to \mathbb{Q}$ defined by evaluation on the diagonal: $$f(S) = \tau(S, S, S)$$ for any $S \in V$. This map $f$ satisfies the homogeneity condition $f(a \cdot S) = a^3 f(S)$ for all scalars $a \in \mathbb{Q}$ and vectors $S \in V$.

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$. Let $\tau$ be a symmetric trilinear form on $V$, and let $f: V \to \mathbb{Q}$ be the associated homogeneous cubic map defined by $f(X) = \tau(X, X, X)$. For any elements $S, T \in V$, the following identity holds: $$f(S + T) = f(S) + f(T) + 3\tau(S, S, T) + 3\tau(T, T, S)$$