内积是将向量标量的函数。

内部产品-$ f：\ mathbb {R} ^ n \ times \ mathbb {R} ^ n \ rightarrow \ kappa $其中$ \ kappa $是标量。

内部产品的基本特征如下-

让$ X \ in \ mathbb {R} ^ n $

• $ \ left \ langle x，x \ right \ rangle \ geq 0，\ forall x \ in X $

• $ \ left \ langle x，x \ right \ rangle = 0 \ Leftrightarrow x = 0，\ forall x \ in X $

• $ \ left \ langle \ alpha x，y \ right \ rangle = \ alpha \ left \ langle x，y \ right \ rangle，\ forall \ alpha \ in \ kappa \：和\：\ forall x，y \ in X $

• $ \ left \ langle x + y，z \ right \ rangle = \ left \ langle x，z \ right \ rangle + \ left \ langle y，z \ right \ rangle，\ forall x，y，z \ in X $

• $ \ left \ langle \ overline {y，x} \ right \ rangle = \ left(x，y \ right)，\ forall x，y \ in X $

注意–

• 规范与内部产品之间的关系：$ \ left \ | x \ right \ | = \ sqrt {\ left(x，x \ right)} $

• $ \ forall x，y \ in \ mathbb {R} ^ n，\ left \ langle x，y \ right \ rangle = x_1y_1 + x_2y_2 + … + x_ny_n $

例子

1.找到$ x = \ left(1,2,1 \ right)\：和\：y = \ left(3，-1,3 \ right)$的内积

解

$ \左\ langle x，y \右\ rangle = x_1y_1 + x_2y_2 + x_3y_3 $

$ \ left \ langle x，y \ right \ rangle = \ left(1 \ times3 \ right)+ \ left(2 \ times-1 \ right)+ \ left(1 \ times3 \ right)$

$ \ left \ langle x，y \ right \ rangle = 3 + \ left(-2 \ right)+ 3 $

$ \左\ langle x，y \右\ rangle = 4 $

2.如果$ x = \ left(4,9,1 \ right)，y = \ left(-3,5,1 \ right)$和$ z = \ left(2,4,1 \ right)$，找到$ \ left(x + y，z \ right)$

解

众所周知，$ \ left \ langle x + y，z \ right \ rangle = \ left \ langle x，z \ right \ rangle + \ left \ langley，z \ right \ rangle $

$ \ left \ langle x + y，z \ right \ rangle = \ left(x_1z_1 + x_2z_2 + x_3z_3 \ right)+ \ left(y_1z_1 + y_2z_2 + y_3z_3 \ right)$

$ \ left \ langle x + y，z \ right \ rangle = \ left \ {\ left(4 \ times 2 \ right)+ \ left(9 \ times 4 \ right)+ \ left(1 \ times1 \ right) \ right \} + $

$ \ left \ {\ left(-3 \ times2 \ right)+ \ left(5 \ times4 \ right)+ \ left(1 \ times 1 \ right)\ right \} $

$ \左\ langle x + y，z \右\ rangle = \左(8 + 36 + 1 \ right)+ \左(-6 + 20 + 1 \ right)$

$ \左\ langle x + y，z \右\ rangle = 45 + 15 $

$ \左\ langle x + y，z \右\ rangle = 60 $