1 . 在平面直角坐标系
中,利用公式
①(其中
,
,
,
为常数),将点
变换为点
的坐标,我们称该变换为线性变换,也称①为坐标变换公式,该变换公式①可由
,
,
,
组成的正方形数表
唯一确定,我们将
称为二阶矩阵,矩阵通常用大写英文字母
,
,…表示.
中,将点
绕原点
按逆时针旋转
得到点
(到原点距离不变),求点
的坐标;
(2)如图,在平面直角坐标系
中,将点
绕原点
按逆时针旋转
角得到点
(到原点距离不变),求坐标变换公式及对应的二阶矩阵;
(3)向量
(称为行向量形式),也可以写成
,这种形式的向量称为列向量,线性变换坐标公式①可以表示为:
,则称
是二阶矩阵
与向量
的乘积,设
是一个二阶矩阵,
,
是平面上的任意两个向量,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba6e18ee381b4e43352acb377fdb4bf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44ee82573986d4fa6a7ee1b5f397edae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0859290725efef72a1b04f473d07da6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c84c4a85e9f31e35bc48c15d9873a03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c84c4a85e9f31e35bc48c15d9873a03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39822cb6df5463c27ac9bfed261a2ff6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d88591679796c52024d11c4de641bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/762bfc20a2da28b3c59225851ea40036.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/762bfc20a2da28b3c59225851ea40036.png)
(2)如图,在平面直角坐标系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44ee82573986d4fa6a7ee1b5f397edae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0859290725efef72a1b04f473d07da6e.png)
(3)向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4224bf1cbcd51f4cbdce93d981d65c5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c17f1f4912527319e32f60e7523c65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c26b9e508047e76f3a7ad88d587702ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd47bfcd685d2466ee27c01bf286406.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c84c4a85e9f31e35bc48c15d9873a03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c17f1f4912527319e32f60e7523c65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f7a1df960feef63dec4790d63f52279.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b8a88a16125366536cb4ad658e0cf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/595e7e1d74355ac82dcfc16b3e86cf78.png)
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7卷引用:安徽省皖江名校联盟2024届高三下学期4月模拟数学试题
安徽省皖江名校联盟2024届高三下学期4月模拟数学试题(已下线)模块五 专题5 全真拔高模拟1(高一人教B版期中)(已下线)数学(新高考卷02,新题型结构)(已下线)模块五 专题5 全真拔高模拟1(苏教版期中研习高一)(已下线)压轴题02圆锥曲线压轴题17题型汇总-1湖南省湘楚名校2023-2024学年高二下学期5月月考数学试题黑龙江省实验中学2024届高三第四次模拟考试数学试题
2 . 对于任意给定的四个实数
,
,
,
,我们定义方阵
,方阵
对应的行列式记为
,且
,方阵
与任意方阵
的乘法运算定义如下:
,其中方阵
,且
.设
,
,
.
(1)证明:
.
(2)若方阵
,
满足
,且
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e84c30444f13d37ada78285dc4f83b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e76d1d8e50dda4d50229a8a20c57e58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc29ee719feeedfbc8c529cf11348abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33e11a5b70e1e2e685d1783a4707872e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4ec97af19b15cd584710a3faf30c716.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f44b167b4e75af29a18637f71f3ebfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b39fcc210ec89dbc7d684a70a34542c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d17ebf9f595cdb9dab841dec703b512.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16a4ed514630bd37fab9765b3fb5f2cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/709d09c76c222f156df31a1bba5f2ce9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2e4a35eca00ea2f4580d62515d54d5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95035eeae686e910be45f08093e406c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93e7d309cb178b71c6e56f5b7f610413.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/109b4ece615b08a89a7f69d436f448b0.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/addb109c49695bce8c5b5cf4fad95772.png)
(2)若方阵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2221c60bc15c59fa1b3ac74a23b57cdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90fa9bfe3bf3e3b7265da3c49d31f1bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35536fb98d8b24cead230c8df95fd9d3.png)
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