Linear Mappings between Vector Spaces form Vector Space

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Theorem

Let $\struct {F, +, \times}$ be a field whose unity is $1_F$.

Let $X, Y$ be vector spaces over the same field $\struct {F, +, \times}$.

Let $\map \LL {X, Y}$ be the set of linear mappings.

Let $x \in X$.

Define pointwise addition $T + S \in \map \LL {X, Y}$ such that:

$\forall x \in X: \map {\paren {T + S} } x := \map T x + \map S x$

For $\alpha \in F$, define pointwise scalar multiplication $\alpha \cdot T$ such that:

$\forall x \in X: \map {\paren {\alpha \cdot T} } x := \alpha \cdot \map T x$

Let $\mathbf 0: X \to Y$ be the zero mapping.


Then $\struct {\map \LL {X, Y}, +, \, \cdot \,}$ is a vector space over $\struct {F, +, \times}$.


Proof

Let $T, S, P \in \map \LL {X, Y}$ such that:

$T, S, P: X \to Y$

Let $\lambda, \mu \in F$.


Vector Space Axiom $\text V 0$: Closure

$\forall x \in X: \map T x, \map S x \in Y$.

By assumption, $Y$ is a vector space.

By Vector Space Axiom $\text V 0$: Closure, $T + S \in \map \LL {X, Y}$

$\Box$


Vector Space Axiom $\text V 1$: Commutativity

$\forall x \in X: \map T x, \map S x \in Y$.

By assumption, $Y$ is a vector space.

By Vector Space Axiom $\text V 1$: Commutativity, $T + S = S + T$

$\Box$


Vector Space Axiom $\text V 2$: Associativity

$\forall x \in X: \map T x, \map S x, \map P x \in Y$.

By assumption, $Y$ is a vector space.

By Vector Space Axiom $\text V 2$: Associativity, $\paren {T + S} + P = T + \paren {S + P}$.

$\Box$


Vector Space Axiom $\text V 3$: Identity

\(\ds \map {\paren {\mathbf 0 + T} } x\) \(=\) \(\ds \map {\mathbf 0} x + \map T x\) Definition of Pointwise Addition of Linear Transformations
\(\ds \) \(=\) \(\ds \mathbf 0_Y + \map T x\) Definition of Zero Mapping
\(\ds \) \(=\) \(\ds \map T x\) Vector Space Axiom $\text V 3$: Identity

$\Box$


Vector Space Axiom $\text V 4$: Inverses

We have that:

$\forall x \in X: \map T x \in Y$

By assumption, $Y$ is a vector space.

By Vector Space Axiom $\text V 4$: Inverses:

$\forall x \in X \exists \paren {- \map T x}: \map T x + \paren {-\map T x} = \mathbf 0_Y$


$\Box$


Vector Space Axiom $\text V 5$: Distributivity over Scalar Addition

\(\ds \map {\paren { \paren {\lambda + \mu} \cdot T} } x\) \(=\) \(\ds \paren {\lambda + \mu} \cdot \map T x\) Definition of Pointwise Scalar Multiplication of Linear Operators
\(\ds \) \(=\) \(\ds \lambda \cdot \map T x + \mu \cdot \map T x\) $\forall x \in X: \map T x \in Y$, Vector Space Axiom $\text V 5$: Distributivity over Scalar Addition
\(\ds \) \(=\) \(\ds \map {\paren {\lambda \cdot T} } x + \map {\paren {\mu\cdot T} } x\) Definition of Pointwise Scalar Multiplication of Linear Operators
\(\ds \) \(=\) \(\ds \map {\paren {\lambda \cdot T + \mu \cdot T} } x\) Definition of Pointwise Addition of Linear Transformations

$\Box$


Vector Space Axiom $\text V 6$: Distributivity over Vector Addition

\(\ds \lambda \cdot \map {\paren {T + S} } x\) \(=\) \(\ds \lambda \cdot \paren {\map T x + \map S x}\) Definition of Pointwise Addition of Linear Transformations
\(\ds \) \(=\) \(\ds \lambda \cdot \map T x + \lambda \cdot \map S x\) $\forall x \in X: \map T x, \map S x \in Y$, Vector Space Axiom $\text V 6$: Distributivity over Vector Addition
\(\ds \) \(=\) \(\ds \map {\paren {\lambda \cdot T} } x + \map {\paren {\lambda \cdot S} } x\) Definition of Pointwise Scalar Multiplication of Linear Operators
\(\ds \) \(=\) \(\ds \map {\paren {\lambda \cdot T + \mu \cdot S} } x\) Definition of Pointwise Addition of Linear Transformations

$\Box$


Vector Space Axiom $\text V 7$: Associativity with Scalar Multiplication

\(\ds \map {\paren {\paren {\lambda \times \mu} \cdot T} } x\) \(=\) \(\ds \paren {\lambda \times \mu} \cdot \map T x\) Definition of Pointwise Scalar Multiplication of Linear Operators
\(\ds \) \(=\) \(\ds \lambda \cdot \paren {\mu \cdot \map T x}\) $\forall x \in X: \map T x \in Y$, Vector Space Axiom $\text V 7$: Associativity with Scalar Multiplication
\(\ds \) \(=\) \(\ds \lambda \cdot \map {\paren {\mu \cdot T} } x\) Definition of Pointwise Scalar Multiplication of Linear Operators
\(\ds \) \(=\) \(\ds \map {\paren {\lambda \cdot \paren {\mu \cdot T} } } x\) Definition of Pointwise Scalar Multiplication of Linear Operators

$\Box$


Vector Space Axiom $\text V 8$: Identity for Scalar Multiplication

\(\ds \map {\paren {1_F \cdot T} } x\) \(=\) \(\ds 1_F \cdot \map T x\) Definition of Pointwise Scalar Multiplication of Linear Operators
\(\ds \) \(=\) \(\ds \map T x\) $\forall x \in X : \map T x \in Y$, Vector Space Axiom $\text V 8$: Identity for Scalar Multiplication

$\blacksquare$


Sources

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