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Diffstat (limited to 'mathematics/algebra.md')
| -rw-r--r-- | mathematics/algebra.md | 7 |
1 files changed, 3 insertions, 4 deletions
diff --git a/mathematics/algebra.md b/mathematics/algebra.md index 1da221b..0ba7847 100644 --- a/mathematics/algebra.md +++ b/mathematics/algebra.md @@ -5,8 +5,7 @@ title: mathematics/algebra # algebra -modern algebra is the study of **algebraic structures**: groups, rings, fields, modules, and the like. -these structures are very abstract: and so results can be applied to a wide variety of situations. +Modern algebra is the study of **algebraic structures**: groups, rings, fields, modules, and the like. These structures are very abstract: and so results can be applied to a wide variety of situations. ## structures @@ -45,8 +44,8 @@ A **vector space** $V$ over a field $F$ of scalars is a set with a binary operat - additive inverse: $∀v, ∃-v: v+(-v) = 0$ - commutativity: $∀u,v : u+v=v+u$ - $(V, )$ is a *scalar operation*: - - scalar identity: $∃1 ∈ F : 1v = v1 = v$ - - commutativity: $∀a,b ∈ F, ∀v ∈ V (ab)v = a(bv)$ + - scalar identity: $∃1 ∈ F, ∀v ∈ V : 1v = v1 = v$ + - commutativity: $∀a,b ∈ F, ∀v ∈ V : (ab)v = a(bv)$ - The *distributive laws* hold: - $∀a ∈ F, ∀u,v ∈ V : a(u+v) = au+av$ - $∀a,b ∈ F, ∀v ∈ V : (a+b)v = av + bv$ |