Physlib.Mathematics.LinearPMap

For any partial linear map $f$, the difference of $f$ with itself is less than or equal to the zero partial linear map, i.e., $f - f \leq 0$. In the context of partial linear maps, the inequality $g \leq h$ means that the domain of $g$ is a subspace of the domain of $h$ and that $g$ and $h$ coincide on the domain of $g$.

For any partial linear map $f$, the sum of its negation $-f$ and itself is less than or equal to the zero partial linear map, denoted as $-f + f \leq 0$. In this context, the relation $g \leq h$ indicates that $h$ is an extension of $g$ (i.e., the domain of $g$ is a subspace of the domain of $h$, and they coincide on the domain of $g$).

For any two partial linear maps $g_1$ and $g_2$, the relation $g_1 \le g_2$ holds if and only if $-g_1 \le -g_2$. Here, the relation $\le$ denotes the extension of partial maps, where $g_1 \le g_2$ means that the domain of $g_1$ is a subspace of the domain of $g_2$ and the maps agree on the domain of $g_1$.

For any two partial linear maps $g_1$ and $g_2$, the relation $g_1 \le -g_2$ holds if and only if $-g_1 \le g_2$ holds. Here, the inequality $f \le g$ denotes that the partial linear map $g$ is an extension of $f$, and $-g$ denotes the pointwise negation of the map.

For any two partial linear maps $f_1$ and $f_2$, the inequality $f_1 + (f_2 - f_1) \le f_2$ holds. In this context, for partial linear maps $g$ and $f$, the relation $g \le f$ signifies that the domain of $g$ is a subspace of the domain of $f$ and $g$ coincides with $f$ on the domain of $g$. The operations of addition and subtraction for partial linear maps are defined on the intersection of the domains of the operands.

For any partial linear maps $f_1$ and $f_2$, the inequality $f_1 + f_2 - f_1 \le f_2$ holds. Here, the inequality $g \le h$ for partial linear maps denotes that the domain of $g$ is a subspace of the domain of $h$ and the two maps agree on the domain of $g$.

For any partial linear maps $f_1$ and $f_2$, it holds that $f_1 + f_2 - f_2 \leq f_1$. In this context, the inequality $f \leq g$ denotes that $g$ is an extension of $f$, meaning the domain of $f$ is a subspace of the domain of $g$ and $f(x) = g(x)$ for all $x$ in the domain of $f$.

For partial linear maps $f_1, f_2, f_3$, the inequality $f_1 + f_2 + (f_3 - f_2) \leq f_1 + f_3$ holds. In the context of partial linear maps, $f \leq g$ means that $g$ is an extension of $f$ (i.e., the domain of $f$ is a subset of the domain of $g$ and the maps agree on the domain of $f$).

For any partial linear maps $f_1, f_2,$ and $f_3$, the following inequality holds: $$f_1 + f_2 - (f_1 - f_3) \le f_2 + f_3$$ where the inequality $f \le g$ denotes that $g$ is an extension of $f$ (i.e., the domain of $f$ is a subspace of the domain of $g$, and $f(x) = g(x)$ for all $x$ in the domain of $f$).

For any three partial linear maps $f_1, f_2,$ and $f_3$, it holds that $(f_1 - f_2) - (f_3 - f_2) \leq f_1 - f_3$. Here, the relation $f \leq g$ between partial linear maps signifies that the domain of $f$ is a subspace of the domain of $g$ and that $f(x) = g(x)$ for all $x$ in the domain of $f$.

For any partial linear maps $f_1$, $f_2$, and $f_3$, the following inequality holds: $$(f_1 - f_2) - (f_1 - f_3) \leq f_3 - f_2$$ where the subtraction $f - g$ of two partial linear maps is defined on the intersection of their domains, and the relation $f \leq g$ denotes that $f$ is a restriction of $g$ (i.e., the domain of $f$ is a subspace of the domain of $g$ and the maps agree on the domain of $f$).

Let $g, g_1$, and $g_2$ be partial linear maps. If $g \le g_1 + g_2$, then $g - g_2 \le g_1$. Here, the inequality $f \le h$ between partial linear maps indicates that the domain of $f$ is a subspace of the domain of $h$ and the maps coincide on the domain of $f$.

For any partial linear maps $f_1$ and $f_2$, the partial linear map $f_1 - f_2 + f_2$ is a restriction of $f_1$, expressed as $f_1 - f_2 + f_2 \leq f_1$.

For any partial linear maps $g, g_1$, and $g_2$, if $g \le g_1 - g_2$, then $g + g_2 \le g_1$. (Here, the relation $\le$ for partial linear maps denotes that the domain of the smaller map is a subspace of the domain of the larger map, and the maps coincide on the smaller domain).

Let $f, g_1,$ and $g_2$ be partial linear maps. If $g_1 \le g_2$, then $f + g_1 \le f + g_2$. The relation $g_1 \le g_2$ for partial linear maps signifies that the domain of $g_1$ is a subspace of the domain of $g_2$ and the maps agree on the domain of $g_1$.

Let $g_1$, $g_2$, and $f$ be partial linear maps. If $g_1 \le g_2$ (meaning the domain of $g_1$ is contained in the domain of $g_2$ and $g_2$ agrees with $g_1$ on the domain of $g_1$), then $g_1 + f \le g_2 + f$.

For any partial linear maps $g_1$, $g_2$, and $f$, if $g_1 \le g_2$, then $g_1 - f \le g_2 - f$. Here, the relation $g_1 \le g_2$ means that the domain of $g_1$ is a subspace of the domain of $g_2$ and $g_1$ is the restriction of $g_2$ to the domain of $g_1$.

For any partial linear maps $f$, $g_1$, and $g_2$, if $g_1 \leq g_2$, then $f - g_1 \leq f - g_2$. Here, the relation $g_1 \leq g_2$ means that $g_2$ is an extension of $g_1$; that is, the domain of $g_1$ is a subspace of the domain of $g_2$ and $g_1(x) = g_2(x)$ for all $x$ in the domain of $g_1$.

For any partial linear map $f : E \to F$ over a scalar field $\mathbb{k}$, the scalar product of $0$ and $f$ is less than or equal to the zero partial linear map: $$(0 : \mathbb{k}) \cdot f \leq 0$$ In the context of partial linear maps, the relation $g \leq h$ means that the domain of $g$ is a subspace of the domain of $h$ and $g$ agrees with $h$ on that subspace.

Let $f$ be a partial linear map from $E$ to $F$ over a field $\mathbb{k}$. If the domain of $f$ is the entire space $E$ (denoted by $\top$), then the scalar multiplication of $f$ by $0 \in \mathbb{k}$ is equal to the zero partial linear map. That is, $0 \cdot f = 0$.

Given a finite family of partial linear maps $\{f_a\}_{a \in \alpha}$ from a module $E$ to $F$ over a ring $R$, the sum $\sum_{a \in \alpha} f_a$ is a partial linear map $E \to F$. Its domain is defined as the intersection of the domains of all maps in the family, $\bigcap_{a \in \alpha} \text{dom}(f_a)$. The value of the sum at an element $x$ in this intersection is the sum of the values of each individual partial linear map, $(\sum_{a \in \alpha} f_a)(x) = \sum_{a \in \alpha} f_a(x)$.

For a finite family of partial linear maps $\{f_a\}_{a \in \alpha}$, the domain of their sum $\sum_{a \in \alpha} f_a$ is the intersection of the domains of the individual maps $f_a$. In mathematical notation, this is expressed as $\text{dom}(\sum_{a \in \alpha} f_a) = \bigcap_{a \in \alpha} \text{dom}(f_a)$.

Given a finite family of partial linear maps $\{f_i\}_{i \in \alpha}$, the domain of their sum $\sum_{i \in \alpha} f_i$ is a submodule of the domain of each individual map $f_a$ for any $a \in \alpha$.

For a finite family of partial linear maps $\{f_a\}_{a \in \alpha}$ and an element $\psi$ belonging to the domain of their sum $\sum_{a \in \alpha} f_a$, the value of the sum evaluated at $\psi$ is the sum of each map evaluated at $\psi$: $$\left(\sum_{a \in \alpha} f_a\right)(\psi) = \sum_{a \in \alpha} f_a(\psi).$$ Note that $\psi$ belongs to the domain of the sum if and only if it belongs to the intersection of the domains of all individual maps $f_a$.

Given partial linear maps $f: E \to F$ and $g: F \to G$ with domains $\text{dom}(f) \subseteq E$ and $\text{dom}(g) \subseteq F$ respectively, their restricted composition $g \circ_r f$ is the partial linear map from $E$ to $G$ whose domain consists of those $x \in \text{dom}(f)$ such that $f(x) \in \text{dom}(g)$. For such $x$, the value is defined as $(g \circ_r f)(x) = g(f(x))$.

The notation $g \circ_r f$ denotes the restricted composition of two partial linear maps $f$ and $g$. If $f$ is a partial linear map from $E$ to $F$ and $g$ is a partial linear map from $F$ to $G$, then $g \circ_r f$ is defined as the partial linear map whose domain consists of those $x$ in the domain of $f$ for which $f(x)$ is in the domain of $g$. The mapping is given by $(g \circ_r f)(x) = g(f(x))$.

Let $f: E \to F$ and $g: F \to G$ be partial linear maps. The domain of their restricted composition $g \circ_r f$ is contained in the domain of $f$, which is expressed as $\text{dom}(g \circ_r f) \subseteq \text{dom}(f)$.

Let $f: E \to F$ and $g: F \to G$ be partial linear maps with domains $\text{dom}(f) \subseteq E$ and $\text{dom}(g) \subseteq F$. Let $f_{\text{fun}}: \text{dom}(f) \to F$ denote the linear map underlying $f$, and let $\iota: \text{dom}(f) \to E$ be the inclusion map. The domain of the restricted composition $g \circ_r f$ is the image under $\iota$ of the preimage of $\text{dom}(g)$ under $f_{\text{fun}}$: $$\text{dom}(g \circ_r f) = \iota(f_{\text{fun}}^{-1}(\text{dom}(g))).$$

Let $u: E \to F$ and $v: F \to G$ be partial linear maps. For any element $x \in E$, $x$ belongs to the domain of the restricted composition $v \circ_r u$ if and only if $x$ belongs to the domain of $u$ and its image $u(x)$ belongs to the domain of $v$.

Let $E, F$, and $G$ be vector spaces over a ring $R$. For partial linear maps $u: E \to F$ and $v: F \to G$, an element $x \in E$ is in the domain of the restricted composition $v \circ_r u$ if and only if there exists $y \in \text{dom}(u)$ such that $x = y$ and there exists $y' \in \text{dom}(v)$ such that $u(y) = y'$.

For partial linear maps $u: E \to F$ and $v: F \to G$, if $x$ is an element of the domain of the restricted composition $v \circ_r u$, then its image $u(x)$ is contained in the domain of $v$.

Let $u: E \to F$ and $v: F \to G$ be partial linear maps between modules over a ring $R$. For any element $x$ in the domain of the restricted composition $v \circ_r u$, the value of the composition is given by $(v \circ_r u)(x) = v(u(x))$.

For any partial linear map $g: F \to G$ between modules $F$ and $G$ over a ring $R$, and for the zero partial linear map $0: E \to F$ (whose domain is the entire module $E$), their restricted composition satisfies $g \circ_r 0 = 0$. Here, $0$ on the right-hand side denotes the zero partial linear map from $E$ to $G$.

Let $E, F, G,$ and $H$ be modules over a ring $R$. For any partial linear maps $f_1 : G \to H$, $f_2 : F \to G$, and $f_3 : E \to F$, the restricted composition $\circ_r$ is associative, meaning $(f_1 \circ_r f_2) \circ_r f_3 = f_1 \circ_r (f_2 \circ_r f_3)$.

Let $E, F$, and $G$ be modules over a ring $R$. Let $u: E \to F$ and $v: F \to G$ be partial linear maps. If for every $x$ in the domain of $u$, the image $u(x)$ is contained in the domain of $v$, then the restricted composition $v \circ_r u$ is equal to the standard composition $v \circ u$ (denoted as `v.comp u h`).

Let $E, F,$ and $G$ be modules over a ring $R$. Let $u: E \to F$ and $v: F \to G$ be partial linear maps. For any submodule $S$ of $E$, let $u|_S$ denote the restriction of $u$ to the subdomain $\text{dom}(u) \cap S$. If the image of $u|_S$ is contained in the domain of $v$ (i.e., $\forall x \in \text{dom}(u|_S), u(x) \in \text{dom}(v)$), then the composition $v \circ (u|_S)$ is less than or equal to the restricted composition $v \circ_r u$. Here, $f \le g$ means that $\text{dom}(f) \subseteq \text{dom}(g)$ and $f(x) = g(x)$ for all $x \in \text{dom}(f)$, and the restricted composition $v \circ_r u$ is the partial linear map with domain $\{x \in \text{dom}(u) \mid u(x) \in \text{dom}(v)\}$.

Let $E, F,$ and $G$ be modules over a ring $R$. Let $f: E \to F$ be a partial linear map, and let $g, g': F \to G$ be partial linear maps. If $g \le g'$, then the restricted compositions satisfy $g \circ_r f \le g' \circ_r f$. Here, $g \le g'$ denotes that the domain of $g$ is a submodule of the domain of $g'$ and $g(x) = g'(x)$ for all $x \in \text{dom}(g)$, and $\circ_r$ denotes the restricted composition.

Let $E, F,$ and $G$ be modules over a ring $R$. Let $g: F \to G$ be a partial linear map, and let $f, f': E \to F$ be partial linear maps. If $f \le f'$, then the restricted compositions satisfy $g \circ_r f \le g \circ_r f'$. Here, the relation $f \le f'$ means that the domain of $f$ is a submodule of the domain of $f'$ and $f(x) = f'(x)$ for all $x \in \text{dom}(f)$, and $\circ_r$ denotes the restricted composition.

Given partial linear maps $f: E \to F$ and $g: F \to G$, the restricted composition of the negation of $g$ with $f$ is equal to the negation of the restricted composition of $g$ with $f$, which is expressed as $(-g) \circ_r f = -(g \circ_r f)$.

Let $f: E \to F$ and $g: F \to G$ be partial linear maps between modules over a ring $R$. The restricted composition of $g$ with the negation of $f$ is equal to the negation of the restricted composition of $g$ and $f$, that is, $g \circ_r (-f) = -(g \circ_r f)$.

For partial linear maps $f: E \to F$ and $g_1, g_2: F \to G$, the restricted composition of the sum $g_1 + g_2$ with $f$ is equal to the sum of the restricted compositions of $g_1$ and $g_2$ with $f$: $$(g_1 + g_2) \circ_r f = g_1 \circ_r f + g_2 \circ_r f$$ where the restricted composition $g \circ_r f$ is the partial linear map with domain $\{x \in \text{dom}(f) \mid f(x) \in \text{dom}(g)\}$, and the sum of two partial linear maps has a domain equal to the intersection of their respective domains.

For partial linear maps $f: E \to F$ and $g_1, g_2: F \to G$, the restricted composition of the difference $g_1 - g_2$ with $f$ is equal to the difference of the restricted compositions: $$(g_1 - g_2) \circ_r f = g_1 \circ_r f - g_2 \circ_r f$$ where the restricted composition $g \circ_r f$ is the partial linear map whose domain is $\{x \in \text{dom}(f) \mid f(x) \in \text{dom}(g)\}$.

For any partial linear maps $f_1, f_2$ from $E$ to $F$ and $g$ from $F$ to $G$, the sum of the restricted compositions $g \circ_r f_1$ and $g \circ_r f_2$ is less than or equal to the restricted composition of $g$ with the sum $f_1 + f_2$: $$g \circ_r f_1 + g \circ_r f_2 \leq g \circ_r (f_1 + f_2)$$ In the context of partial linear maps, the inequality $L \leq M$ means that the domain of $L$ is a subset of the domain of $M$, and the maps agree on the domain of $L$.

For any partial linear maps $f_1, f_2$ from $E$ to $F$ and $g$ from $F$ to $G$, the difference of the restricted compositions of $g$ with $f_1$ and $f_2$ is less than or equal to the restricted composition of $g$ with the difference $f_1 - f_2$: $$g \circ_r f_1 - g \circ_r f_2 \le g \circ_r (f_1 - f_2)$$ where the inequality $A \le B$ between partial linear maps signifies that $A$ is a restriction of $B$ (i.e., $\text{dom}(A) \subseteq \text{dom}(B)$ and $A(x) = B(x)$ for all $x \in \text{dom}(A)$).

Let $S$ be a division ring and $E, F, G$ be modules over $S$. Given partial linear maps $f: E \to F$ and $g: F \to G$, and a non-zero scalar $c \in S$, the restricted composition of $g$ with the scalar multiple $c \cdot f$ is equal to the scalar multiple of the restricted composition of $g$ with $f$. That is, $$g \circ_r (c \cdot f) = c \cdot (g \circ_r f),$$ where the restricted composition $h \circ_r k$ of two partial linear maps is defined on the domain $\{x \in \text{dom}(k) \mid k(x) \in \text{dom}(h)\}$.

Let $M$ be a monoid and $E, F, G$ be modules over a ring $R$. Suppose $M$ has a distributive action on $G$ that commutes with the action of $R$. For any scalar $c \in M$ and any partial linear maps $f: E \to F$ and $g: F \to G$, the restricted composition of $c \cdot g$ with $f$ is equal to the scalar multiple of the restricted composition of $g$ with $f$: $$(c \cdot g) \circ_r f = c \cdot (g \circ_r f),$$ where the restricted composition $h \circ_r k$ is defined on the domain $\{x \in \text{dom}(k) \mid k(x) \in \text{dom}(h)\}$.

Let $E$ be a module over a ring $R$. The set of partial linear maps from $E$ to itself forms a monoid. The multiplication operation is the restricted composition $g \circ_r f$, whose domain is $\{x \in \text{dom}(f) \mid f(x) \in \text{dom}(g)\}$, and the identity element is the identity linear map $\text{id}_E$ defined on the entire space $E$.

For any two partial linear maps $f_1, f_2$ from a module $E$ to itself, their product $f_1 * f_2$ in the monoid of partial linear maps is equal to their restricted composition $f_1 \circ_r f_2$. The restricted composition is defined on the domain $\{x \in \text{dom}(f_2) \mid f_2(x) \in \text{dom}(f_1)\}$.