CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-11

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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${End}_{F} (V)$ is a ring and matrix representation is a ring isomorphism ${End}_{F} (V) ≅ M_{n} (F)$

Statement

Let $V$ be an $n$ -dimensional vector space over $F$ . Then ${End}_{F} (V) : = L (V, V)$ is a ring under pointwise addition and composition, and for every ordered basis $B$ the map

$T ⟼ [T]_{B}^{B}$

is a ring isomorphism ${End}_{F} (V) ≅ M_{n} (F)$ .

Facts & Assumptions

Given: A finite-dimensional $F$ -vector space $V$ and an ordered basis $B$ of length $n$ .

[L1]

$L (V, V)$ is a vector space under pointwise operations ( $L (V, W)$ is a vector space over the common scalar field).

[L2]

Matrix representation is a vector-space isomorphism and sends composition to matrix multiplication ( $T \mapsto [T]_{B}^{C}$ is a vector-space isomorphism $L (V, W) ≅ M_{m \times n} (F)$ , $[S \circ T]_{B}^{D} = [S]_{C}^{D} [T]_{B}^{C}$ ).

Proof

technique · direct

1.1

Composition of endomorphisms is associative, has ${id}_{V}$ as identity, and distributes over pointwise addition; together with the additive group from [L1], this makes ${End}_{F} (V)$ a ring.

givenL1

2.1

By [L2], matrix representation is a bijective linear map, so it preserves addition and zero.

step 1.1L1L2

3.1

It preserves products by the composition formula in [L2], and $[{id}_{V}]_{B}^{B} = I_{n}$ by coordinate action. Thus it is a bijective unital ring homomorphism and hence a ring isomorphism.

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

S. Axler, Linear Algebra Done Right, 4th ed., §3C (standard reference, not scraped)

End⁡F(V)EndF​(V) is a ring and matrix representation is a ring isomorphism End⁡F(V)≅Mn(F)EndF​(V)≅Mn​(F)