[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.84,0:00:03.08,Default,,0000,0000,0000,,假设已知两个非0的向量
Dialogue: 0,0:00:03.11,0:00:05.16,Default,,0000,0000,0000,,其中一个向量是x
Dialogue: 0,0:00:05.18,0:00:06.92,Default,,0000,0000,0000,,另一个向量是y
Dialogue: 0,0:00:06.95,0:00:13.71,Default,,0000,0000,0000,,它们都在空间Rn中且非0
Dialogue: 0,0:00:16.80,0:00:21.93,Default,,0000,0000,0000,,从而可以看出它们的绝对值――
Dialogue: 0,0:00:21.95,0:00:23.36,Default,,0000,0000,0000,,我换一种颜色
Dialogue: 0,0:00:23.38,0:00:26.44,Default,,0000,0000,0000,,这个颜色比较好
Dialogue: 0,0:00:26.47,0:00:29.94,Default,,0000,0000,0000,,这两个向量的点积的
Dialogue: 0,0:00:29.96,0:00:31.83,Default,,0000,0000,0000,,绝对值――
Dialogue: 0,0:00:31.85,0:00:34.69,Default,,0000,0000,0000,,注意 这个结果是一个标量
Dialogue: 0,0:00:34.72,0:00:39.84,Default,,0000,0000,0000,,它小于等于二者长度的乘积
Dialogue: 0,0:00:39.87,0:00:42.60,Default,,0000,0000,0000,,我们已经定义了点积
Dialogue: 0,0:00:42.63,0:00:43.99,Default,,0000,0000,0000,,也定义了长度
Dialogue: 0,0:00:44.02,0:00:46.57,Default,,0000,0000,0000,,它小于等于二者长度的乘积
Dialogue: 0,0:00:46.59,0:00:48.77,Default,,0000,0000,0000,,进一步地
Dialogue: 0,0:00:48.79,0:00:51.14,Default,,0000,0000,0000,,等号成立的唯一条件
Dialogue: 0,0:00:51.17,0:00:57.09,Default,,0000,0000,0000,,这两个向量的点积
Dialogue: 0,0:00:57.12,0:01:00.48,Default,,0000,0000,0000,,等于它们长度之积的唯一情况是――
Dialogue: 0,0:01:00.52,0:01:02.18,Default,,0000,0000,0000,,等号成立的
Dialogue: 0,0:01:02.20,0:01:04.74,Default,,0000,0000,0000,,唯一条件是――
Dialogue: 0,0:01:04.76,0:01:06.19,Default,,0000,0000,0000,,我写下来――
Dialogue: 0,0:01:06.22,0:01:10.30,Default,,0000,0000,0000,,即其中一个向量可以看做
Dialogue: 0,0:01:10.32,0:01:12.01,Default,,0000,0000,0000,,另一个向量的常数倍
Dialogue: 0,0:01:12.03,0:01:13.39,Default,,0000,0000,0000,,也就是说它们是共线的
Dialogue: 0,0:01:13.42,0:01:16.19,Default,,0000,0000,0000,,即其中一个向量
Dialogue: 0,0:01:16.22,0:01:17.35,Default,,0000,0000,0000,,是另一个向量的伸长或缩短
Dialogue: 0,0:01:17.37,0:01:21.21,Default,,0000,0000,0000,,当且仅当向量x
Dialogue: 0,0:01:21.23,0:01:24.98,Default,,0000,0000,0000,,等于向量y的常数倍时
Dialogue: 0,0:01:25.01,0:01:29.97,Default,,0000,0000,0000,,我们把
Dialogue: 0,0:01:29.99,0:01:31.43,Default,,0000,0000,0000,,这个不等式
Dialogue: 0,0:01:31.46,0:01:33.55,Default,,0000,0000,0000,,称作Cauchy-Schwarz不等式
Dialogue: 0,0:01:43.52,0:01:45.45,Default,,0000,0000,0000,,我们来证明它
Dialogue: 0,0:01:45.48,0:01:46.67,Default,,0000,0000,0000,,因为你不能仅从表面理解
Dialogue: 0,0:01:46.69,0:01:48.60,Default,,0000,0000,0000,,我能单纯地接受它
Dialogue: 0,0:01:48.63,0:01:52.52,Default,,0000,0000,0000,,下面我要构造一个函数
Dialogue: 0,0:01:52.54,0:01:55.05,Default,,0000,0000,0000,,构造一个函数――
Dialogue: 0,0:01:55.09,0:02:00.40,Default,,0000,0000,0000,,它是变量t的函数
Dialogue: 0,0:02:00.42,0:02:02.95,Default,,0000,0000,0000,,定义函数p(t)等于
Dialogue: 0,0:02:02.99,0:02:09.97,Default,,0000,0000,0000,,等于某个向量的长度
Dialogue: 0,0:02:10.00,0:02:15.43,Default,,0000,0000,0000,,这个向量是ty-x 其中t是标量
Dialogue: 0,0:02:15.46,0:02:17.82,Default,,0000,0000,0000,,就是这个向量的长度
Dialogue: 0,0:02:17.85,0:02:18.88,Default,,0000,0000,0000,,这是一个向量
Dialogue: 0,0:02:18.92,0:02:20.40,Default,,0000,0000,0000,,再加一个平方
Dialogue: 0,0:02:20.42,0:02:22.36,Default,,0000,0000,0000,,在继续进行之前
Dialogue: 0,0:02:22.39,0:02:24.01,Default,,0000,0000,0000,,我还要强调一点
Dialogue: 0,0:02:24.03,0:02:26.82,Default,,0000,0000,0000,,如果要取一个向量的长度 我写在这
Dialogue: 0,0:02:26.84,0:02:32.29,Default,,0000,0000,0000,,若要取向量v的长度
Dialogue: 0,0:02:32.30,0:02:34.75,Default,,0000,0000,0000,,我希望大家知道
Dialogue: 0,0:02:34.77,0:02:37.40,Default,,0000,0000,0000,,这是一个正数
Dialogue: 0,0:02:37.43,0:02:39.15,Default,,0000,0000,0000,,至少是大于等于0的
Dialogue: 0,0:02:39.17,0:02:41.84,Default,,0000,0000,0000,,因为这里是平方和
Dialogue: 0,0:02:41.87,0:02:44.98,Default,,0000,0000,0000,,这是v2? 一直到vn?
Dialogue: 0,0:02:45.00,0:02:46.76,Default,,0000,0000,0000,,所有这些都是实数
Dialogue: 0,0:02:46.78,0:02:47.97,Default,,0000,0000,0000,,当对一个实数平方时
Dialogue: 0,0:02:48.00,0:02:50.61,Default,,0000,0000,0000,,结果总是大于等于0的
Dialogue: 0,0:02:50.63,0:02:51.74,Default,,0000,0000,0000,,当把它们加起来时
Dialogue: 0,0:02:51.76,0:02:52.92,Default,,0000,0000,0000,,得到的结果
Dialogue: 0,0:02:52.94,0:02:53.97,Default,,0000,0000,0000,,当然也大于等于0
Dialogue: 0,0:02:53.98,0:02:55.11,Default,,0000,0000,0000,,在对其开根号
Dialogue: 0,0:02:55.13,0:02:56.97,Default,,0000,0000,0000,,取主平方根 即正的平方根
Dialogue: 0,0:02:56.99,0:02:58.03,Default,,0000,0000,0000,,得到的结果
Dialogue: 0,0:02:58.04,0:02:59.19,Default,,0000,0000,0000,,大于等于0
Dialogue: 0,0:02:59.20,0:03:02.44,Default,,0000,0000,0000,,所以任何实向量的长度
Dialogue: 0,0:03:02.46,0:03:03.80,Default,,0000,0000,0000,,都大于等于0
Dialogue: 0,0:03:03.83,0:03:06.72,Default,,0000,0000,0000,,这就是实向量的长度
Dialogue: 0,0:03:06.75,0:03:10.48,Default,,0000,0000,0000,,它大于等于0
Dialogue: 0,0:03:10.51,0:03:12.75,Default,,0000,0000,0000,,在之前的视频中
Dialogue: 0,0:03:12.77,0:03:13.99,Default,,0000,0000,0000,,应该是上上个视频
Dialogue: 0,0:03:14.01,0:03:17.42,Default,,0000,0000,0000,,我也讲了
Dialogue: 0,0:03:17.45,0:03:20.36,Default,,0000,0000,0000,,一个向量的长度的平方可以写成
Dialogue: 0,0:03:20.38,0:03:24.42,Default,,0000,0000,0000,,该向量与自身做点积
Dialogue: 0,0:03:24.45,0:03:26.75,Default,,0000,0000,0000,,我重新写一下这个向量
Dialogue: 0,0:03:26.77,0:03:32.40,Default,,0000,0000,0000,,这个向量长度的平方等于
Dialogue: 0,0:03:32.42,0:03:34.49,Default,,0000,0000,0000,,那个向量与自身的点积
Dialogue: 0,0:03:34.52,0:03:43.09,Default,,0000,0000,0000,,从而等于(ty-x)・(ty-x)
Dialogue: 0,0:03:43.11,0:03:45.98,Default,,0000,0000,0000,,在上个视频中
Dialogue: 0,0:03:46.01,0:03:50.44,Default,,0000,0000,0000,,我告诉了大家
Dialogue: 0,0:03:50.46,0:03:52.38,Default,,0000,0000,0000,,你可以把向量的点积
Dialogue: 0,0:03:52.41,0:03:53.85,Default,,0000,0000,0000,,近似理解为常数的乘积
Dialogue: 0,0:03:53.88,0:03:55.88,Default,,0000,0000,0000,,它们的结合律
Dialogue: 0,0:03:55.91,0:03:58.00,Default,,0000,0000,0000,,分配律和交换律是类似的
Dialogue: 0,0:03:58.02,0:04:00.19,Default,,0000,0000,0000,,所以当对其做乘法时
Dialogue: 0,0:04:00.22,0:04:01.28,Default,,0000,0000,0000,,你可以把它看做
Dialogue: 0,0:04:01.31,0:04:02.36,Default,,0000,0000,0000,,两个二项式相乘
Dialogue: 0,0:04:02.38,0:04:04.46,Default,,0000,0000,0000,,你可以按照常规的代数二项式乘法的方法
Dialogue: 0,0:04:04.47,0:04:07.23,Default,,0000,0000,0000,,来计算这个点积
Dialogue: 0,0:04:07.26,0:04:09.26,Default,,0000,0000,0000,,用到的就是分配律
Dialogue: 0,0:04:09.29,0:04:13.46,Default,,0000,0000,0000,,但要注意 这同常数的乘法还是有区别的
Dialogue: 0,0:04:13.49,0:04:15.78,Default,,0000,0000,0000,,我们做的是点积
Dialogue: 0,0:04:15.80,0:04:17.18,Default,,0000,0000,0000,,是向量的乘法
Dialogue: 0,0:04:17.21,0:04:18.64,Default,,0000,0000,0000,,或者说是向量的一类乘法
Dialogue: 0,0:04:18.66,0:04:20.62,Default,,0000,0000,0000,,把括号打开
Dialogue: 0,0:04:20.64,0:04:24.87,Default,,0000,0000,0000,,有ty・ty
Dialogue: 0,0:04:24.88,0:04:26.06,Default,,0000,0000,0000,,我写下来
Dialogue: 0,0:04:26.09,0:04:29.60,Default,,0000,0000,0000,,这是ty・ty
Dialogue: 0,0:04:29.61,0:04:33.74,Default,,0000,0000,0000,,然后得到――
Dialogue: 0,0:04:33.76,0:04:36.47,Default,,0000,0000,0000,,我这么做
Dialogue: 0,0:04:36.50,0:04:40.82,Default,,0000,0000,0000,,然后得到-x・ty
Dialogue: 0,0:04:40.84,0:04:43.35,Default,,0000,0000,0000,,我不说“乘以”
Dialogue: 0,0:04:43.37,0:04:45.38,Default,,0000,0000,0000,,而应该说是“点乘”
Dialogue: 0,0:04:45.41,0:04:50.46,Default,,0000,0000,0000,,这是x・ty
Dialogue: 0,0:04:50.49,0:04:58.16,Default,,0000,0000,0000,,然后有ty・(-x)
Dialogue: 0,0:04:58.17,0:05:04.25,Default,,0000,0000,0000,,然后是-ty・x
Dialogue: 0,0:05:04.27,0:05:08.28,Default,,0000,0000,0000,,最后一项是x・x
Dialogue: 0,0:05:08.30,0:05:11.94,Default,,0000,0000,0000,,它可以看做是(-1x)・(-1x)
Dialogue: 0,0:05:11.96,0:05:15.83,Default,,0000,0000,0000,,可以写成+(-1x)
Dialogue: 0,0:05:15.85,0:05:21.59,Default,,0000,0000,0000,,这可以看做是加上-1
Dialogue: 0,0:05:21.62,0:05:25.99,Default,,0000,0000,0000,,所以这是(-1x)・(-1x)
Dialogue: 0,0:05:26.02,0:05:27.68,Default,,0000,0000,0000,,我们看一下
Dialogue: 0,0:05:27.71,0:05:28.80,Default,,0000,0000,0000,,这就是上式
Dialogue: 0,0:05:28.83,0:05:31.36,Default,,0000,0000,0000,,展开后的形式
Dialogue: 0,0:05:31.38,0:05:32.67,Default,,0000,0000,0000,,这其实不能算作化简
Dialogue: 0,0:05:32.69,0:05:35.18,Default,,0000,0000,0000,,但是我可以用交换律和分配律
Dialogue: 0,0:05:35.20,0:05:38.30,Default,,0000,0000,0000,,来改写这个表达式
Dialogue: 0,0:05:38.33,0:05:45.34,Default,,0000,0000,0000,,它等于(y・y)t?
Dialogue: 0,0:05:45.36,0:05:46.60,Default,,0000,0000,0000,,t是一个标量
Dialogue: 0,0:05:46.63,0:05:51.40,Default,,0000,0000,0000,,减去―― 事实上应该是2倍的
Dialogue: 0,0:05:51.42,0:05:52.80,Default,,0000,0000,0000,,这两项是相等的
Dialogue: 0,0:05:52.83,0:05:54.72,Default,,0000,0000,0000,,它们仅仅是排列不一样
Dialogue: 0,0:05:54.75,0:05:56.88,Default,,0000,0000,0000,,我们已经知道点积满足结合律
Dialogue: 0,0:05:56.90,0:06:05.32,Default,,0000,0000,0000,,所以等于2(x・y)t
Dialogue: 0,0:06:05.34,0:06:08.91,Default,,0000,0000,0000,,也许我应该换一个颜色
Dialogue: 0,0:06:08.93,0:06:12.72,Default,,0000,0000,0000,,从而这两项就化成这一项
Dialogue: 0,0:06:12.76,0:06:16.11,Default,,0000,0000,0000,,然后如果重新排列这一项
Dialogue: 0,0:06:16.12,0:06:17.50,Default,,0000,0000,0000,,会得到-1乘以-1
Dialogue: 0,0:06:17.52,0:06:19.75,Default,,0000,0000,0000,,它们消去了 从而符号是+
Dialogue: 0,0:06:19.79,0:06:24.59,Default,,0000,0000,0000,,并且仅剩下x・x
Dialogue: 0,0:06:24.62,0:06:26.79,Default,,0000,0000,0000,,我换个颜色
Dialogue: 0,0:06:26.82,0:06:29.31,Default,,0000,0000,0000,,用橘黄色吧
Dialogue: 0,0:06:29.34,0:06:32.57,Default,,0000,0000,0000,,从而这项就化成了这项
Dialogue: 0,0:06:32.59,0:06:35.59,Default,,0000,0000,0000,,当然 这项化成了这项
Dialogue: 0,0:06:35.61,0:06:37.03,Default,,0000,0000,0000,,注意 我所做的
Dialogue: 0,0:06:37.05,0:06:38.77,Default,,0000,0000,0000,,就是改写这个式子
Dialogue: 0,0:06:38.79,0:06:40.34,Default,,0000,0000,0000,,这项大于等于0
Dialogue: 0,0:06:40.36,0:06:43.32,Default,,0000,0000,0000,,我把它写在这
Dialogue: 0,0:06:43.34,0:06:46.28,Default,,0000,0000,0000,,这两个式子是相等的
Dialogue: 0,0:06:46.30,0:06:47.52,Default,,0000,0000,0000,,我只是改写了一下
Dialogue: 0,0:06:47.55,0:06:51.92,Default,,0000,0000,0000,,所以这个式子大于等于0
Dialogue: 0,0:06:51.96,0:06:54.20,Default,,0000,0000,0000,,下面要做一个替换
Dialogue: 0,0:06:54.23,0:06:56.43,Default,,0000,0000,0000,,来化简这个表达式
Dialogue: 0,0:06:56.46,0:06:59.01,Default,,0000,0000,0000,,然后再反替换回来
Dialogue: 0,0:06:59.04,0:07:01.59,Default,,0000,0000,0000,,将它定义为a
Dialogue: 0,0:07:01.62,0:07:08.01,Default,,0000,0000,0000,,定义这一项是b
Dialogue: 0,0:07:08.04,0:07:10.62,Default,,0000,0000,0000,,就是-2x・y这项
Dialogue: 0,0:07:10.65,0:07:11.90,Default,,0000,0000,0000,,保留t
Dialogue: 0,0:07:11.92,0:07:13.82,Default,,0000,0000,0000,,定义这项
Dialogue: 0,0:07:13.84,0:07:18.01,Default,,0000,0000,0000,,将它定义为c
Dialogue: 0,0:07:18.03,0:07:19.67,Default,,0000,0000,0000,,即x・x=c
Dialogue: 0,0:07:19.70,0:07:22.15,Default,,0000,0000,0000,,那么这个表达式化成了什么呢？
Dialogue: 0,0:07:22.17,0:07:28.59,Default,,0000,0000,0000,,它化成at?减去――
Dialogue: 0,0:07:28.60,0:07:30.74,Default,,0000,0000,0000,,我要注意颜色的使用――
Dialogue: 0,0:07:30.77,0:07:35.30,Default,,0000,0000,0000,,然后是bt+c
Dialogue: 0,0:07:35.32,0:07:40.67,Default,,0000,0000,0000,,我们当然知道
Dialogue: 0,0:07:40.68,0:07:42.01,Default,,0000,0000,0000,,它大于等于0
Dialogue: 0,0:07:42.03,0:07:43.74,Default,,0000,0000,0000,,它与上面这项相等
Dialogue: 0,0:07:43.76,0:07:44.87,Default,,0000,0000,0000,,都大于等于0
Dialogue: 0,0:07:44.89,0:07:46.29,Default,,0000,0000,0000,,我把p(t)写在这
Dialogue: 0,0:07:46.31,0:07:48.97,Default,,0000,0000,0000,,这项现在大于等于0
Dialogue: 0,0:07:49.00,0:07:51.71,Default,,0000,0000,0000,,对于任意的t成立
Dialogue: 0,0:07:51.73,0:07:54.03,Default,,0000,0000,0000,,对于任给的实数t
Dialogue: 0,0:07:54.05,0:08:04.84,Default,,0000,0000,0000,,我取函数在b/2a处的值
Dialogue: 0,0:08:04.86,0:08:07.33,Default,,0000,0000,0000,,我确定可以这么做
Dialogue: 0,0:08:07.36,0:08:08.89,Default,,0000,0000,0000,,因为我确定
Dialogue: 0,0:08:08.90,0:08:10.36,Default,,0000,0000,0000,,分母上不会出现0
Dialogue: 0,0:08:10.38,0:08:13.90,Default,,0000,0000,0000,,a是这个向量与自身做点积
Dialogue: 0,0:08:13.91,0:08:16.08,Default,,0000,0000,0000,,并且已知它是非0的向量
Dialogue: 0,0:08:16.11,0:08:18.71,Default,,0000,0000,0000,,它是这个向量长度的平方
Dialogue: 0,0:08:18.73,0:08:20.19,Default,,0000,0000,0000,,它是非0向量
Dialogue: 0,0:08:20.20,0:08:21.68,Default,,0000,0000,0000,,对于上面这些项
Dialogue: 0,0:08:21.71,0:08:23.03,Default,,0000,0000,0000,,当你取其长度时
Dialogue: 0,0:08:23.04,0:08:24.26,Default,,0000,0000,0000,,结果都是正的
Dialogue: 0,0:08:24.28,0:08:26.04,Default,,0000,0000,0000,,所以这一项是非0的
Dialogue: 0,0:08:26.06,0:08:27.23,Default,,0000,0000,0000,,它是非0的向量
Dialogue: 0,0:08:27.25,0:08:30.48,Default,,0000,0000,0000,,从而2乘以这个点积
Dialogue: 0,0:08:30.50,0:08:31.55,Default,,0000,0000,0000,,也是非0的
Dialogue: 0,0:08:31.58,0:08:32.74,Default,,0000,0000,0000,,所以我们可以这么做
Dialogue: 0,0:08:32.76,0:08:34.86,Default,,0000,0000,0000,,不用担心除以0的事
Dialogue: 0,0:08:34.89,0:08:37.03,Default,,0000,0000,0000,,它等于什么呢？
Dialogue: 0,0:08:37.05,0:08:38.25,Default,,0000,0000,0000,,它等于――
Dialogue: 0,0:08:38.26,0:08:39.35,Default,,0000,0000,0000,,我还用绿色来写
Dialogue: 0,0:08:39.37,0:08:41.82,Default,,0000,0000,0000,,要换颜色太麻烦了
Dialogue: 0,0:08:41.85,0:08:44.55,Default,,0000,0000,0000,,它等于a乘以这个表达式的平方
Dialogue: 0,0:08:44.58,0:08:48.37,Default,,0000,0000,0000,,即b?/4a?
Dialogue: 0,0:08:48.39,0:08:51.64,Default,,0000,0000,0000,,将2a平方得到4a?
Dialogue: 0,0:08:51.66,0:08:55.52,Default,,0000,0000,0000,,减去b乘以这项
Dialogue: 0,0:08:55.54,0:08:58.97,Default,,0000,0000,0000,,即b乘以） 这是常数的乘法
Dialogue: 0,0:08:58.99,0:09:01.64,Default,,0000,0000,0000,,即b乘以b/2a
Dialogue: 0,0:09:01.66,0:09:03.47,Default,,0000,0000,0000,,这是常数的乘法
Dialogue: 0,0:09:03.49,0:09:05.04,Default,,0000,0000,0000,,再加上c
Dialogue: 0,0:09:05.06,0:09:07.35,Default,,0000,0000,0000,,我们这道这个式子大于等于0
Dialogue: 0,0:09:07.38,0:09:11.62,Default,,0000,0000,0000,,如果将其化简 得到什么呢？
Dialogue: 0,0:09:11.65,0:09:15.46,Default,,0000,0000,0000,,这里的a消去了
Dialogue: 0,0:09:15.48,0:09:18.22,Default,,0000,0000,0000,,分子上是b?
Dialogue: 0,0:09:18.25,0:09:26.07,Default,,0000,0000,0000,,从而有b?/4a-b?/2a
Dialogue: 0,0:09:26.09,0:09:27.53,Default,,0000,0000,0000,,就是这一项
Dialogue: 0,0:09:27.56,0:09:30.93,Default,,0000,0000,0000,,再加上c 这个这项大于等于0
Dialogue: 0,0:09:30.96,0:09:32.77,Default,,0000,0000,0000,,我改写一下
Dialogue: 0,0:09:32.79,0:09:35.27,Default,,0000,0000,0000,,如果分子分母同时乘以2
Dialogue: 0,0:09:35.29,0:09:37.72,Default,,0000,0000,0000,,会得到什么？
Dialogue: 0,0:09:37.74,0:09:41.08,Default,,0000,0000,0000,,得到2b?/4a
Dialogue: 0,0:09:41.10,0:09:42.55,Default,,0000,0000,0000,,我这么做是因为
Dialogue: 0,0:09:42.57,0:09:43.84,Default,,0000,0000,0000,,我要使分母相同
Dialogue: 0,0:09:43.86,0:09:45.59,Default,,0000,0000,0000,,那么得到什么？
Dialogue: 0,0:09:45.63,0:09:49.07,Default,,0000,0000,0000,,得到b?/4a-2b?/4a
Dialogue: 0,0:09:49.09,0:09:52.07,Default,,0000,0000,0000,,化简之后是什么？
Dialogue: 0,0:09:52.08,0:09:55.38,Default,,0000,0000,0000,,分子是b?-2b?
Dialogue: 0,0:09:55.40,0:10:00.63,Default,,0000,0000,0000,,从而就得到-b?/4a+c
Dialogue: 0,0:10:00.65,0:10:02.63,Default,,0000,0000,0000,,大于等于0
Dialogue: 0,0:10:02.66,0:10:06.14,Default,,0000,0000,0000,,这两项相加得到这项
Dialogue: 0,0:10:06.16,0:10:09.27,Default,,0000,0000,0000,,如果在等式两边加上这项
Dialogue: 0,0:10:09.28,0:10:16.19,Default,,0000,0000,0000,,得到c大于等于b?/4a
Dialogue: 0,0:10:16.23,0:10:17.82,Default,,0000,0000,0000,,这项在左边是负的
Dialogue: 0,0:10:17.85,0:10:19.07,Default,,0000,0000,0000,,如果在两边同时加上它
Dialogue: 0,0:10:19.10,0:10:20.57,Default,,0000,0000,0000,,则右边的项就变成正的
Dialogue: 0,0:10:20.59,0:10:22.91,Default,,0000,0000,0000,,我们得到的东西
Dialogue: 0,0:10:22.94,0:10:24.64,Default,,0000,0000,0000,,是一个不等式
Dialogue: 0,0:10:24.67,0:10:28.31,Default,,0000,0000,0000,,现在把变量替换回去
Dialogue: 0,0:10:28.34,0:10:29.74,Default,,0000,0000,0000,,看看得到什么
Dialogue: 0,0:10:29.77,0:10:32.75,Default,,0000,0000,0000,,我开始做的替换在哪？
Dialogue: 0,0:10:32.77,0:10:34.12,Default,,0000,0000,0000,,它在这
Dialogue: 0,0:10:34.14,0:10:37.28,Default,,0000,0000,0000,,进一步化简
Dialogue: 0,0:10:37.32,0:10:40.06,Default,,0000,0000,0000,,两边同时乘以4a
Dialogue: 0,0:10:40.08,0:10:43.42,Default,,0000,0000,0000,,a不仅是非0的
Dialogue: 0,0:10:43.45,0:10:44.54,Default,,0000,0000,0000,,而且是正的
Dialogue: 0,0:10:44.56,0:10:45.97,Default,,0000,0000,0000,,这是它长度的平方
Dialogue: 0,0:10:46.00,0:10:47.77,Default,,0000,0000,0000,,并且我已经讲过
Dialogue: 0,0:10:47.80,0:10:50.96,Default,,0000,0000,0000,,任何实向量的长度都是正的
Dialogue: 0,0:10:50.99,0:10:52.83,Default,,0000,0000,0000,,我之所以要强调
Dialogue: 0,0:10:52.85,0:10:54.58,Default,,0000,0000,0000,,a是正的是因为
Dialogue: 0,0:10:54.60,0:10:55.82,Default,,0000,0000,0000,,如果两边同时乘以它
Dialogue: 0,0:10:55.84,0:10:57.26,Default,,0000,0000,0000,,不等式就不用变号
Dialogue: 0,0:10:57.28,0:10:59.30,Default,,0000,0000,0000,,那么在做替换之前
Dialogue: 0,0:10:59.33,0:11:00.61,Default,,0000,0000,0000,,我在两边同时乘以a
Dialogue: 0,0:11:00.63,0:11:07.20,Default,,0000,0000,0000,,得到4ac大于等于b?
Dialogue: 0,0:11:07.23,0:11:08.54,Default,,0000,0000,0000,,得到这个
Dialogue: 0,0:11:08.58,0:11:10.27,Default,,0000,0000,0000,,我煞费苦心地做到了这一步
Dialogue: 0,0:11:10.30,0:11:12.84,Default,,0000,0000,0000,,我说过a一定是正数
Dialogue: 0,0:11:12.86,0:11:15.34,Default,,0000,0000,0000,,因为它是向量长度的平方
Dialogue: 0,0:11:15.36,0:11:17.75,Default,,0000,0000,0000,,y・y是y的长度的平方
Dialogue: 0,0:11:17.77,0:11:19.58,Default,,0000,0000,0000,,它是一个正值
Dialogue: 0,0:11:19.60,0:11:20.81,Default,,0000,0000,0000,,它一定是正的
Dialogue: 0,0:11:20.83,0:11:21.97,Default,,0000,0000,0000,,我们在实数范围内处理问题
Dialogue: 0,0:11:22.00,0:11:23.75,Default,,0000,0000,0000,,现在来做替换
Dialogue: 0,0:11:23.77,0:11:29.41,Default,,0000,0000,0000,,那么4a就是y・y
Dialogue: 0,0:11:29.44,0:11:31.47,Default,,0000,0000,0000,,y・y也是――
Dialogue: 0,0:11:31.51,0:11:33.17,Default,,0000,0000,0000,,我还写在这
Dialogue: 0,0:11:33.19,0:11:39.25,Default,,0000,0000,0000,,y・y就是y的长度的平方
Dialogue: 0,0:11:39.27,0:11:41.32,Default,,0000,0000,0000,,这是y・y 它等于a
Dialogue: 0,0:11:41.35,0:11:45.17,Default,,0000,0000,0000,,我在之前的视频中讲过y・y
Dialogue: 0,0:11:45.19,0:11:46.65,Default,,0000,0000,0000,,乘以c
Dialogue: 0,0:11:46.68,0:11:47.99,Default,,0000,0000,0000,,c是x・x
Dialogue: 0,0:11:48.01,0:11:52.58,Default,,0000,0000,0000,,x・x就等于
Dialogue: 0,0:11:52.60,0:11:55.28,Default,,0000,0000,0000,,向量x的长度的平方
Dialogue: 0,0:11:55.31,0:11:57.43,Default,,0000,0000,0000,,这是c
Dialogue: 0,0:11:57.45,0:12:00.54,Default,,0000,0000,0000,,从而4ac
Dialogue: 0,0:12:00.56,0:12:03.59,Default,,0000,0000,0000,,大于等于b?
Dialogue: 0,0:12:03.62,0:12:05.55,Default,,0000,0000,0000,,那么b是多少？ b就在这里
Dialogue: 0,0:12:05.58,0:12:14.63,Default,,0000,0000,0000,,从而b?等于2(x・y)?
Dialogue: 0,0:12:14.66,0:12:17.29,Default,,0000,0000,0000,,我们得到了这个结果
Dialogue: 0,0:12:17.30,0:12:19.53,Default,,0000,0000,0000,,我们下面怎么做呢？
Dialogue: 0,0:12:19.56,0:12:21.44,Default,,0000,0000,0000,,抱歉 应该是对整个这项平方
Dialogue: 0,0:12:21.47,0:12:22.93,Default,,0000,0000,0000,,这一项才是b
Dialogue: 0,0:12:22.96,0:12:24.61,Default,,0000,0000,0000,,我们看看能否进行化简
Dialogue: 0,0:12:24.64,0:12:27.36,Default,,0000,0000,0000,,我们得到―― 我换一种颜色
Dialogue: 0,0:12:27.39,0:12:32.46,Default,,0000,0000,0000,,4乘以y的长度的平方
Dialogue: 0,0:12:32.49,0:12:35.73,Default,,0000,0000,0000,,乘以x的长度的平方
Dialogue: 0,0:12:35.76,0:12:37.34,Default,,0000,0000,0000,,大于等于――
Dialogue: 0,0:12:37.36,0:12:38.70,Default,,0000,0000,0000,,如果对这项平方
Dialogue: 0,0:12:38.72,0:12:44.61,Default,,0000,0000,0000,,就得到4(x・y)
Dialogue: 0,0:12:44.63,0:12:53.62,Default,,0000,0000,0000,,再乘以(x・y)
Dialogue: 0,0:12:53.65,0:12:55.63,Default,,0000,0000,0000,,事实上
Dialogue: 0,0:12:55.66,0:12:56.83,Default,,0000,0000,0000,,这么写会更好一些
Dialogue: 0,0:12:56.84,0:13:00.57,Default,,0000,0000,0000,,写成4(x・y)?
Dialogue: 0,0:13:00.59,0:13:02.92,Default,,0000,0000,0000,,下面两边同时除以4
Dialogue: 0,0:13:02.94,0:13:04.50,Default,,0000,0000,0000,,这不会改变不等式
Dialogue: 0,0:13:04.51,0:13:06.06,Default,,0000,0000,0000,,两边的4就消去了
Dialogue: 0,0:13:06.09,0:13:07.97,Default,,0000,0000,0000,,现在对等式两边
Dialogue: 0,0:13:07.99,0:13:09.22,Default,,0000,0000,0000,,同时开平方
Dialogue: 0,0:13:09.25,0:13:12.67,Default,,0000,0000,0000,,则两端开平方之后得――
Dialogue: 0,0:13:12.69,0:13:14.10,Default,,0000,0000,0000,,这些都是正值
Dialogue: 0,0:13:14.12,0:13:15.45,Default,,0000,0000,0000,,所以这边开平方
Dialogue: 0,0:13:15.47,0:13:16.93,Default,,0000,0000,0000,,就是每一项的开方
Dialogue: 0,0:13:16.95,0:13:18.76,Default,,0000,0000,0000,,这是根据指数的性质
Dialogue: 0,0:13:18.79,0:13:20.34,Default,,0000,0000,0000,,如果对两边开方
Dialogue: 0,0:13:20.36,0:13:25.74,Default,,0000,0000,0000,,就得到y的长度乘以x的长度
Dialogue: 0,0:13:25.75,0:13:30.02,Default,,0000,0000,0000,,大于等于这一项的开方
Dialogue: 0,0:13:30.04,0:13:31.46,Default,,0000,0000,0000,,我们开方后取正值
Dialogue: 0,0:13:31.49,0:13:33.33,Default,,0000,0000,0000,,不等式两边开方后
Dialogue: 0,0:13:33.36,0:13:34.49,Default,,0000,0000,0000,,都取正值
Dialogue: 0,0:13:34.51,0:13:36.44,Default,,0000,0000,0000,,这使得我们
Dialogue: 0,0:13:36.45,0:13:37.98,Default,,0000,0000,0000,,避免了许多麻烦
Dialogue: 0,0:13:38.00,0:13:40.22,Default,,0000,0000,0000,,从而正的平方根
Dialogue: 0,0:13:40.24,0:13:44.26,Default,,0000,0000,0000,,就是x・y的绝对值
Dialogue: 0,0:13:44.29,0:13:45.73,Default,,0000,0000,0000,,严格地说
Dialogue: 0,0:13:45.77,0:13:46.79,Default,,0000,0000,0000,,这个是绝对值
Dialogue: 0,0:13:46.82,0:13:48.08,Default,,0000,0000,0000,,因为这一项
Dialogue: 0,0:13:48.11,0:13:52.11,Default,,0000,0000,0000,,可能是负值
Dialogue: 0,0:13:52.13,0:13:53.17,Default,,0000,0000,0000,,但当平方之后
Dialogue: 0,0:13:53.20,0:13:55.67,Default,,0000,0000,0000,,你应当注意
Dialogue: 0,0:13:55.69,0:13:56.78,Default,,0000,0000,0000,,当开方之后
Dialogue: 0,0:13:56.79,0:13:58.07,Default,,0000,0000,0000,,还保持正值
Dialogue: 0,0:13:58.10,0:14:01.99,Default,,0000,0000,0000,,否则的话 当我们取主平方根后
Dialogue: 0,0:14:02.01,0:14:03.27,Default,,0000,0000,0000,,就会产生混乱
Dialogue: 0,0:14:03.30,0:14:06.48,Default,,0000,0000,0000,,我们取正的平方根
Dialogue: 0,0:14:06.52,0:14:07.53,Default,,0000,0000,0000,,就是――
Dialogue: 0,0:14:07.57,0:14:08.67,Default,,0000,0000,0000,,如果取绝对值
Dialogue: 0,0:14:08.70,0:14:11.34,Default,,0000,0000,0000,,就确定了值是正的
Dialogue: 0,0:14:11.37,0:14:12.39,Default,,0000,0000,0000,,这是我们的结果
Dialogue: 0,0:14:12.42,0:14:14.87,Default,,0000,0000,0000,,向量点积的绝对值
Dialogue: 0,0:14:14.90,0:14:19.54,Default,,0000,0000,0000,,小于等于这两个向量长度的乘积
Dialogue: 0,0:14:19.57,0:14:20.64,Default,,0000,0000,0000,,从而就证明了
Dialogue: 0,0:14:20.65,0:14:21.95,Default,,0000,0000,0000,,Cauchy-Schwarz不等式
Dialogue: 0,0:14:27.53,0:14:29.83,Default,,0000,0000,0000,,我最后要说的是
Dialogue: 0,0:14:29.86,0:14:37.29,Default,,0000,0000,0000,,如果x是y的常数倍
Dialogue: 0,0:14:37.30,0:14:39.13,Default,,0000,0000,0000,,会怎么样？
Dialogue: 0,0:14:39.15,0:14:41.82,Default,,0000,0000,0000,,如果这样的话 绝对值是多少？
Dialogue: 0,0:14:41.85,0:14:45.20,Default,,0000,0000,0000,,x・y的绝对值
Dialogue: 0,0:14:45.23,0:14:48.82,Default,,0000,0000,0000,,它等于―― 等于什么？
Dialogue: 0,0:14:48.84,0:14:49.88,Default,,0000,0000,0000,,如果进行替换
Dialogue: 0,0:14:49.92,0:14:53.01,Default,,0000,0000,0000,,则它等于cy・y的绝对值
Dialogue: 0,0:14:53.03,0:14:54.88,Default,,0000,0000,0000,,它就是x・y
Dialogue: 0,0:14:54.92,0:14:57.54,Default,,0000,0000,0000,,从而等于
Dialogue: 0,0:14:57.57,0:15:00.31,Default,,0000,0000,0000,,根据结合律
Dialogue: 0,0:15:00.35,0:15:03.81,Default,,0000,0000,0000,,它等于绝对值――
Dialogue: 0,0:15:03.84,0:15:06.13,Default,,0000,0000,0000,,我们已经确定
Dialogue: 0,0:15:06.16,0:15:07.62,Default,,0000,0000,0000,,绝对值总是正的
Dialogue: 0,0:15:07.64,0:15:10.50,Default,,0000,0000,0000,,然后是y・y
Dialogue: 0,0:15:10.51,0:15:20.60,Default,,0000,0000,0000,,这等于c乘以y的长度――
Dialogue: 0,0:15:20.66,0:15:23.95,Default,,0000,0000,0000,,等于y的长度的平方
Dialogue: 0,0:15:23.98,0:15:30.52,Default,,0000,0000,0000,,从而等于c的大小――
Dialogue: 0,0:15:30.54,0:15:32.83,Default,,0000,0000,0000,,或者说常数c的绝对值
Dialogue: 0,0:15:32.84,0:15:35.20,Default,,0000,0000,0000,,乘以向量y的长度
Dialogue: 0,0:15:39.17,0:15:43.93,Default,,0000,0000,0000,,我可以改写这一项
Dialogue: 0,0:15:43.96,0:15:46.13,Default,,0000,0000,0000,,如果你对其不确定的话
Dialogue: 0,0:15:46.16,0:15:48.19,Default,,0000,0000,0000,,可以自己证明一下 但是这一项――
Dialogue: 0,0:15:48.22,0:15:50.19,Default,,0000,0000,0000,,我们可以把c放入绝对值中
Dialogue: 0,0:15:50.22,0:15:51.83,Default,,0000,0000,0000,,这是个很好的证明练习
Dialogue: 0,0:15:51.86,0:15:53.11,Default,,0000,0000,0000,,直接证明就可以
Dialogue: 0,0:15:53.14,0:15:54.26,Default,,0000,0000,0000,,只需应用长度的定义
Dialogue: 0,0:15:54.28,0:15:55.66,Default,,0000,0000,0000,,并且用它乘以c
Dialogue: 0,0:15:55.68,0:16:00.82,Default,,0000,0000,0000,,从而等于cy的长度乘以――
Dialogue: 0,0:16:00.85,0:16:04.53,Default,,0000,0000,0000,,我应该说cy的长度乘以y的长度
Dialogue: 0,0:16:04.55,0:16:09.00,Default,,0000,0000,0000,,有几个向量没有标上记号
Dialogue: 0,0:16:09.03,0:16:11.36,Default,,0000,0000,0000,,我给它们标上
Dialogue: 0,0:16:11.38,0:16:13.08,Default,,0000,0000,0000,,这是x
Dialogue: 0,0:16:13.11,0:16:19.15,Default,,0000,0000,0000,,所以它等于x的长度乘以y的长度
Dialogue: 0,0:16:19.18,0:16:21.54,Default,,0000,0000,0000,,我给大家介绍了
Dialogue: 0,0:16:21.55,0:16:23.08,Default,,0000,0000,0000,,Cauchy-Schwarz不等式的第二部分
Dialogue: 0,0:16:23.10,0:16:25.13,Default,,0000,0000,0000,,式子两边相等
Dialogue: 0,0:16:25.16,0:16:28.22,Default,,0000,0000,0000,,当且仅当它们互为对方的常数倍
Dialogue: 0,0:16:28.24,0:16:29.47,Default,,0000,0000,0000,,如果你对我讲的某些步骤
Dialogue: 0,0:16:29.48,0:16:31.04,Default,,0000,0000,0000,,还有些疑惑
Dialogue: 0,0:16:31.07,0:16:32.54,Default,,0000,0000,0000,,那么你可以去证明一下
Dialogue: 0,0:16:32.55,0:16:35.04,Default,,0000,0000,0000,,例如 证明||cy||
Dialogue: 0,0:16:35.07,0:16:38.88,Default,,0000,0000,0000,,与|c|*||y||
Dialogue: 0,0:16:38.90,0:16:41.55,Default,,0000,0000,0000,,是相等的
Dialogue: 0,0:16:41.57,0:16:43.98,Default,,0000,0000,0000,,无论如何 我希望你能发觉它的用处
Dialogue: 0,0:16:44.00,0:16:46.13,Default,,0000,0000,0000,,在证明线性代数中的一些结论时
Dialogue: 0,0:16:46.16,0:16:47.40,Default,,0000,0000,0000,,我们经常会
Dialogue: 0,0:16:47.44,0:16:49.51,Default,,0000,0000,0000,,用到Cauchy-Schwarz不等式
Dialogue: 0,0:16:49.53,0:16:50.63,Default,,0000,0000,0000,,在下面的视频中
Dialogue: 0,0:16:50.64,0:16:51.97,Default,,0000,0000,0000,,我会更直观地为大家讲解
Dialogue: 0,0:16:52.00,0:16:53.05,Default,,0000,0000,0000,,为什么这个不等式
Dialogue: 0,0:16:53.07,0:16:54.72,Default,,0000,0000,0000,,与向量的点积有关