章节大纲

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    In earlier lessons we learned about the dot product in  R n  however, now we want to generalize that notion to any sort of vector space  V .
    ::然而,在早先我们学到的Rn圆点产品的经验教训中,我们现在想将这一概念推广到任何类型的矢量空间V。


    An inner product denoted by the tuple  ,  satisfied the four  essential properties that
    ::满足了四种基本特性,

    1. u + v , w = u , w + v , w 2. α v , w = α v , w 3. v , w = w , v 4. v , v 0  if and only if  v = 0
    ::万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万万

    Here are some examples of inner products
    ::以下是一些内部产品的例子。

    1. In the real numbers we can define the inner product as  x , y = x y
    ::1. 以实际数字,我们可以将内产物定义为xx,yxy

    2. In  C [ a , b ] , the set of all continuous functions from  a  to  b  we have the inner product of  f , g = a b f ( x ) g ( x )  d x
    ::2. 在C[a,b]中,从a到b的所有连续函数组,我们的内产物为f,gabf(x)g(x)dx

    Finally, in this lesson we'll mainly focus on the dot product which in  R n  has the form  x 1 , , x n , y 1 , , y n = x 1 y 1 + + x n y n
    ::最后,在这个课程中,我们将主要关注在Rn的圆点产品,其形式是x1,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,。

    We have that the length of any vector is  | | x | | = x , x  we also get that | | c v | | = | c | | | v | |
    ::任何矢量的长度为 xxxxxxxxxxxcvcv

    We can also get that the distance between  u  and  v  is  dist ( u v ) = u v , u v
    ::我们还可以理解,u和v之间的距离是分离的(uv)uv,uv


    Now let's talk about orthogonality.
    ::现在,让我们来谈谈垂直论。

    Two vectors are orthogonal if their dot product is 0.
    ::如果两个矢量的点产值为 0, 则两个矢量是正向的 。

    Then, we also get  u , v  are orthogonal if and only if  u + v , u + v = u + v
    ::然后,我们得到u,v是正统的,如果,并且只有如果,uv,uvuvv


     

    Now, let's look at the orthogonal complement of a vector.
    ::现在,让我们看看矢量的正对角补充。

    We denote the orthogonal complement of a vector space as  W  if  W  is a vector space.
    ::如果W是矢量空间,我们表示矢量空间的正向补充是W。

    This is the orthogonal complement if the following properties hold:
    ::如果有下列属性持有, 这就是正向补全 :

    1. A vector  x  is in  W  if and only if  x  is orthogonal to every vector in a set that spans  W .
    ::1. 矢量 x 在 W 中,如果且仅在 x 与每个矢量的正对值在W 中,则该矢量在W 中。

    2.  W  is a subspace of  R n .
    ::2. W是Rn的子空间。

    Here is a theorem about orthogonal complements that is important to understand:
    ::以下是关于正向补充的理论,

    Let A be an m by n matrix. The orthogonal complement of the row space of A is the null space of A and the orthogonal complement of the column space of A is the null space of the transpose of A. In other words:
    ::Let A be an by n 矩阵。 A 行空间的正对角补充是 A 的空格, A 列空间的正对角补充是 A 转换的空格。 换句话说:

    ( Row  A ) = Nul  A ( Col  A ) = Nul  A T
    :伤心Row A) Nul A(Col A) Nul AT

    Proof:
    ::证据:

    When computing the matrix product  A x  we see that if  x Nul  A  then  x  is orthogonal to each row of  A .  Because the rows of A when taken the span of form the row space our vector is orthogonal to Row A. Going in the other direction, if x is orthogonal to the row space of A then x  must be orthogonal to each row of A and thus Ax = 0 proving our first statement. A similar argument proves the second statement of the theorem. [Try it on your own as an exercise]
    ::当计算矩阵产品 Ax 时,我们看到如果 xNul A 然后 x 与 A 的每行是正对的。因为 A 行的行在形式间距上是正对的。因为 A 行的行在形式间距上是正对的,我们的矢量在行间是正对的。朝另一个方向走,如果 x 是正对的, 那么 x 在 A 的 行间距上必须是正对的, 因此A 行的每行必须是正对的, 因此 Ax = 0 证明了我们的第一个语句。 类似的论点证明了该词的第二个语句 。 [用练习本身尝试它]


    Here are two videos to help you develop your understanding on this topic.
    ::以下是两段影片, 帮助您了解这个话题。

     


     

    Problems:
    ::问题:

    1. Calculate the inner product:  1 , 3 , 2 , 1 , 4 , 8 = [ 1 3 2 ] [ 1 4 8 ]
    ::1. 计算内产物:%1,-3,2,%1,4,8[1-32][-148]

    2. Prove this statement or salvage to make true in the case that it's false: In a square matrix (n by n) the vectors in the column space of that matrix are orthogonal to the vectors in the  null space of that matrix.
    ::2. 证明这一声明或救助在虚假的情况下成为事实:在平方矩阵(n x n)中,该矩阵列空间的矢量与该矩阵空格中的矢量正对。

    3. Find the orthogonal complement of  Span { [ 4 2 1 3 ] }
    ::3. 查找斯潘{[4-213]} 的正向补全{[4-213]}

    4. List 3 examples of inner products and their corresponding vector spaces.
    ::4. 列举3个内部产品及其相应的矢量空间的例子。