Isomorphism 同构

小结：

1、两个有限维度的向量空间，在同一数域下，是同构的 等价于 它们维数相等。

Isomorphism 同构

0.1.8 Isomorphism. If U and V are vector spaces over the same scalar field F, and if f : U → V is an invertible function such that f (ax + by) = a f (x) + bf (y) for all x, y ∈ U and all a, b ∈ F, then f is said to be an isomorphism and U and V are said to be isomorphic (“same structure”). Two finite-dimensional vector spaces over the same field are isomorphic if and only if they have the same dimension; thus, any n-dimensional vector space over F is isomorphic to Fn. Any n-dimensional real vector space is, therefore, isomorphic to Rn, and any n-dimensional complex vector space is isomorphic to Cn. Specifically, if V is an n-dimensional vector space over a field F 0.2 Matrices 5 with specified basis B = {x1,..., xn}, then, since any element x ∈ V may be written uniquely as x = a1x1 +···+ an xn in which each ai ∈ F, we may identify x with the nvector [x]B = [a1 ... an] T . For any basis B, the mapping x → [x]B is an isomorphism between V and Fn.