2.1.6 向量空间

有维向量的集合，如果在其上定义了加法运算和数乘运算，且对两种运算封闭，即运算结果仍属于此集合，则称为向量空间，也称线性空间。对于任意的向量,都有

则集合为向量空间。根据线性组合的定义，向量空间中任意向量的线性组合仍然属于此空间。
向量空间的极大线性无关组称为空间的基，基所包含的向量数称为空间的维数。
如果基向量相互正交

则称为正交基。如果向量基向量相互正交且长度均为1

则称为标准正交基。
给定一组线性无关的向量，可以根据它们构造出标准正交基，采用的方法是格拉姆-施密特正交化。给定一组非0且线性无关的向量，格拉姆-施密特正交化先构造出一组正交基,然后将这组正交基进行标准化。首先选择向量做为第一个正交基，令

然后加入,构造和的线性组合，使得它与正交

由于与正交，因此有

解得

接下来加入，构造出,是和的线性组合，使得它与及均正交

由于与正交，因此有

而与正交，,因此可以解得
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同理可得
以此类推，在加入时构造下面的线性组合
由于它与均正交，因此
而与, 均正交，从而解得