显示数据的矩阵

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Introduction
::导言

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A matrix is a rectangular array of numbers representing data in a variety of forms. Computers work very heavily with matrices because operations with matrices are efficient with memory. Matrices can represent statistical data with numbers, but also graphical data with pictures. 
::矩阵是一个代表不同形式数据的矩形数字阵列,计算机对矩阵作用很大,因为矩阵操作对内存有效。矩阵可以代表带有数字的统计数据,也可以代表带有图片的图形数据。

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How might you use a matrix to write the image below as something a computer could recognize and work with?
::您如何使用矩阵来写下下面的图像, 作为计算机能够识别和工作的东西?

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A matrix is a means of storing information effectively and efficiently. The rows and columns each mean something very specific, and the location of a number is just as important as its value. The following are all examples of matrices:
::矩阵是有效和高效地储存信息的一种手段。行和列每个都意味着非常具体的东西,数字的位置与其价值同样重要。

$\begin{array}{r}\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right],\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\left[\begin{array}{cccc}2& 3& 4& 5\\ 0& 4& 5& 0\\ 0& 0& 2& -9\\ 0& 0& 0& 10\end{array}\right]\end{array}$

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The entries in a matrix can be written out using brackets like [ ], but they can also be described individually using a set of two subscript indices,  $\begin{array}{r}i\end{array}$  and $\begin{array}{r}j,\end{array}$  which stand for the row number and the column number. Alternatively, the matrix can be named with just a capital letter like $\begin{array}{r}A\end{array}$ .
::矩阵中的条目可以用括号如[ ]来写出,但也可以用一组两条下标指数(i和j),即行号和列号,分别加以说明。 或者,矩阵可以用象A这样的大写字母命名。

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$\begin{array}{r}A=[a_{ij}]=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\end{array}$
::A=[aij]=[a11a12a21a22]

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Square matrices have the same number of rows as columns. The order of a matrix , or the  dimensions of a matrix , describes the number of rows and the number of columns in the matrix. The matrix below is said to have order $\begin{array}{r}2\times 3\end{array}$  because it has two rows and three columns. A $\begin{array}{r}1\times 1\end{array}$  matrix is just a regular number.
::方块矩阵的行数与列数相同。矩阵的顺序或矩阵的尺寸说明矩阵中的行数和列数。以下的矩阵据说有顺序2×3,因为它有两个行和三列。A 1×1矩阵只是一个普通数字。

$\begin{array}{r}\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]\end{array}$

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The identity matrix of order $\begin{array}{r}n\times n\end{array}$  has zeros everywhere, except along the main diagonal where it has ones. The identity matrix of any order has special properties. The following are examples of identity matrices of order 1, 2, and 3, respectively:
::nxn顺序的身份矩阵除主要对角有零外,无处不在。任何顺序的身份矩阵都有特殊属性。以下分别是第1、2和3顺序的身份矩阵实例:

$\begin{array}{r}[1],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}$

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A symmetric matrix is a special type of square matrix that has reflection across the main diagonal. The identity matrix is an example of a symmetric matrix.
::对称矩阵是一种特殊的平方矩阵类型,在主对角反射,身份矩阵是对称矩阵的一个例子。

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When you turn the rows of a matrix into the columns of a new matrix, the two matrices are transpositions of one another. The superscript $\begin{array}{r}T\end{array}$  stands for transpose. Sometimes using the transpose of a matrix is more useful than using the matrix itself.
::当您将矩阵的行转换为新矩阵的列时,两个矩阵是相互转换的。上标 T 代表转换。有时,使用矩阵的转换比使用矩阵本身更有用。

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::A=[123456]AT=[142536]

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A triangular matrix is a matrix that has a triangle of zeros within the matrix. A   lower triangular matrix is a square matrix where every entry below the diagonal is zero. An upper triangular matrix is a square matrix where every entry above the diagonal is zero. Shown below is a lower triangular matrix. When you work with solving matrices, look for triangular matrices because they are much easier to solve.
::三角矩阵是一个矩阵,在矩阵内有一个零的三角形。下三角矩阵是一个方格矩阵,在对角线下每个条目为零。上三角矩阵是一个方格矩阵,在对角线上每个条目为零。下面显示的是一个较低三角矩阵。当您在解析矩阵时,寻找三角矩阵,因为三角矩阵更容易解答。

$\begin{array}{r}\left[\begin{array}{cccc}2& 3& 4& 5\\ 0& 4& 5& 0\\ 0& 0& 2& -9\\ 0& 0& 0& 10\end{array}\right]\end{array}$

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A diagonal matrix is both upper and lower triangular, which means all the entries except those along the diagonal are zero. The identity matrix is a special case of a diagonal matrix. 
::对角矩阵为上三角形和下三角形,这意味着除对角矩阵外的所有条目均为零。身份矩阵是对角矩阵的一个特例。

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The following video further  explains how to determine the dimensions of a matrix, and why it is important to be able to determine the dimensions of a matrix: 
::以下视频进一步解释了如何确定矩阵的维度,以及为何必须能够确定矩阵的维度:

 

 

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Play, Learn, and Explore Matrices:  .
::玩耍、学习和探索母体:.

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Examples
::实例

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Example 1
::例1

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In the Introduction, you were asked how you might use a matrix to write the image below as something a computer could recognize and work with. 
::在导言中,有人问您如何使用矩阵书写下面的图像,作为计算机能够识别和工作的东西。

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Solution:
::解决方案 :

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By writing every hollow square as a 0 and every blank square as a 1, a computer could read the picture.
::将每个空方块写成0,将每个空方块写成1,计算机就能读出图片。

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When you use computers to manipulate images, the computer manipulates just the numbers. In this case, if you swap zeros and ones, you get the negative image.
::当您使用计算机操作图像时, 计算机只操作数字。 在这种情况下, 如果您交换零和一, 您就会得到负图像 。

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Real photos and computer images have matrices that are much larger and include more numbers than just zero and one to account for more colors.
::真实照片和计算机图像的矩阵要大得多,包含的数字多于零和一,以说明更多颜色。

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Example 2
::例2

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Kate runs three bakeries, and each bakery sells bagels and muffins. 
::凯特经营三个面包店 每个面包店都卖百吉饼和松饼

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The rows represent the bakeries, and the columns represent bagels (left) and muffins (right) sold. 
::各行代表面包店,各栏代表面包圈(左)和松饼(右)出售的面包圈。

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::K=[1441921151272734]

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Answer the questions below about Kate's sales.
::回答以下关于凯特销售的问题。

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1) What does 127 represent?
::1) 127代表什么?

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Solution:
::解决方案 :

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It represents the number of muffins Kate sold in her 2nd location. You know this because it is in the muffin column and the 2nd row.
::它代表了凯特在第二处销售的松饼数量。你知道这一点,因为它在松饼栏和第二排。

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2) How many muffins did Kate sell in total?
::2)凯特总共卖了多少松饼?

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Solution:
::解决方案 :

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The total muffins sold is equal to the sum of the righthand column:   $\begin{array}{r}192+127+34=353\end{array}$ .
::销售的松饼总数等于右栏总和:192+127+34=353。

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3) How many bagels did Kate sell at her 1st location?
:3) Kate在第一地点卖了多少个百吉饼?

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Solution:
::解决方案 :

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Kate sold 144 bagels at her 1st location.
::Kate在第一地点卖了144个百吉饼

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4) Which location is doing poorly?
::4) 哪个地点情况不佳?

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Solution:
::解决方案 :

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The 3rd location is doing much worse than the other two locations.
::第三地点比其他两个地点差得多

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Example 3
::例3

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Identify the order of the following matrices:
::确定下列矩阵的顺序:

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::A=[1347],B=[214513415],C=[25235556256224413434331141]

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Solution:
::解决方案 :

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$\begin{array}{r}A\end{array}$ is $\begin{array}{r}1\times 4\end{array}$ , $\begin{array}{r}B\end{array}$  is $\begin{array}{r}2\times 3\end{array}$ , $\begin{array}{r}C\end{array}$  is $\begin{array}{r}5\times 2\end{array}$ .  Note that  $\begin{array}{r}4\times 1,3\times 2,2\times 5\end{array}$ are not the same orders and would be incorrect. 
::A 是 1x4, B 是 2x3, C 是 5x2. 注意 4x1, 1x2, 2x5 并非同一订单, 且不正确 。

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Example 4
::例4

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Write out the $\begin{array}{r}5\times 4\end{array}$  matrix whose entries are $\begin{array}{r}{a}_{ij}=\frac{i+j}{j}\end{array}$ .
::填写 Aij=i+jj 条目的 5x4 矩阵。

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Solution:
::解决方案 :

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Example 5
::例5

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Create a $\begin{array}{r}3\times 3\end{array}$  matrix for each of the following:
::为以下每一种创建 3x3 矩阵 :

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1) Diagonal Matrix
:1) 对角矩阵

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Solution:
::解决方案 :

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Possible answer:
::可能的答复:

  $\begin{array}{r}\left[\begin{array}{ccc}4& 0& 0\\ 0& 3& 0\\ 0& 0& 5\end{array}\right]\end{array}$

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2) Lower Triangular Matrix
:2) 下三角矩阵

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Solution:
::解决方案 :

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Possible answer:
::可能的答复:

$\begin{array}{r}\left[\begin{array}{ccc}4& 1& 1\\ 0& 3& 14\\ 0& 0& 5\end{array}\right]\end{array}$

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3) Symmetric Matrix
::3) 对称矩阵

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Solution:
::解决方案 :

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Possible answer:
::可能的答复:

$\begin{array}{r}\left[\begin{array}{ccc}4& 1& 1\\ 1& 3& 14\\ 1& 14& 5\end{array}\right]\end{array}$

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4) Identity Matrix
:4) 身份矩阵

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Solution:
::解决方案 :

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Possible Answer:
::可能的答复:

$\begin{array}{r}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}$

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Note:  While the identity matrix is  technically a correct answer for all four parts of this problem, it does not highlight the differences between each definition.
::注:虽然身份信息总库在技术上是对该问题所有四个部分的正确答案,但它没有突出每个定义之间的差异。

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Summary
::摘要

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  A matrix is a rectangular array of numbers representing data.
  ::矩阵是一个代表数据的矩形数字阵列。

  
  ::矩阵是一个代表数据的矩形数字阵列。
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  Square matrices have the same number of rows as columns. 
  ::方格矩阵的行数与列数相同。

  
  ::方格矩阵的行数与列数相同。
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  The order  or dimension of a matrix describes the number of rows and the number of columns in the matrix. 
  ::矩阵的顺序或尺寸说明矩阵中的行数和列数。

  
  ::矩阵的顺序或尺寸说明矩阵中的行数和列数。
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  A symmetric matrix is a special type of square matrix that has reflection symmetry across the main diagonal. The identity matrix is an example of a symmetric matrix.
  ::对称矩阵是一种特殊的平方矩阵类型,在主对角上反射对称。身份矩阵是对称矩阵的一个实例。

  
  ::对称矩阵是一种特殊的平方矩阵类型,在主对角上反射对称。身份矩阵是对称矩阵的一个实例。
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  The identity matrix of order $\begin{array}{r}n\times n\end{array}$  has zeros everywhere, except along the main diagonal where it has ones. Just like the number one has an important property with numbers, the identity matrix of any order has special properties as well.
  ::排序 nxn 的身份矩阵无处不在, 但主要对角有零。 正如排名第一的属性与数字一样, 任何顺序的身份矩阵也有特殊属性 。

  
  ::排序 nxn 的身份矩阵无处不在, 但主要对角有零。 正如排名第一的属性与数字一样, 任何顺序的身份矩阵也有特殊属性 。

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State the order of each of the following matrices:
::列出下列表格的顺序:

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1.
     $\begin{array}{r}A=\left[\begin{array}{cccc}4& 2& 4& 7\\ 5& 2& 1& 0\end{array}\right]\end{array}$
::1. A=[42475210]

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2.
     $\begin{array}{r}B=\left[\begin{array}{cc}0& 1\\ 34& 1\end{array}\right]\end{array}$
::2. B=[01341]

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3.
     $\begin{array}{r}C=\left[\begin{array}{cc}2& 62\\ 14& 3\\ 4& 3\\ 1& 11\end{array}\right]\end{array}$
::3. C=[2624143111]

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4.
    $\begin{array}{r}D=\left[\begin{array}{ccc}12& 0& 2\\ 0& 3& 3\\ 4& 0& 1\\ 1& 4& 0\end{array}\right]\end{array}$
::4. D=[12033401140]

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5.   $\begin{array}{r}E=\left[\begin{array}{cc}1& 11\end{array}\right]\end{array}$
::5. E=[111]

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6. Give an example of a $\begin{array}{r}1\times 1\end{array}$  matrix.
::6. 举例说明1x1矩阵。

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7. Give an example of a $\begin{array}{r}3\times 2\end{array}$  matrix.
::7. 举一个3x2矩阵的例子。

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8. If a symmetric matrix is also lower triangular, what type of matrix is it?
::8. 如果对称矩阵也是较低三角形的,那么它是什么类型的矩阵?

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9. Write out the $\begin{array}{r}2\times 3\end{array}$  matrix whose entries are $\begin{array}{r}{a}_{ij}=i-j\end{array}$ . 
::9. 填写条目为aij=i-j的2×3矩阵。

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Morgan worked for three weeks during the summer, earning money on Mondays, Tuesdays, Wednesdays, Thursdays, and Fridays. The following matrix represents his earnings:
::Morgan在夏季工作了三周,在星期一、星期二、星期三、星期四和星期五挣钱。

$\begin{array}{r}\left[\begin{array}{ccc}24& 22& 32\\ 25& 28& 30\\ 30& 28& 32\\ 10& 15& 19\\ 35& 32& 30\end{array}\right]\end{array}$

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10. What do the rows and columns represent?
::10. 列和列代表什么?

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11. How much money did Morgan make in the 1st week?
::11. Morgan一周赚了多少钱?

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12. How much money did Morgan make on Tuesdays?
::12. 摩根星期二赚了多少钱?

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13. What day of the week was most profitable?
::13. 星期的哪一天最有利可图?

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14. What day of the week was least profitable?
::14. 星期的哪一天利润最低?

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15. Is the following a matrix? Explain.
::15. 以下是矩阵表吗?

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$\begin{array}{r}\left[\begin{array}{cc}\text{dogs}& 0\\ \text{cats}& 3\\ \text{sheep}& 0\\ \text{ducks}& 4\end{array}\right]\end{array}$
::[狗猫3she0ducks4] [狗猫3she0ducks4]

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Review (Answers)
::回顾(答复)

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Please see the Appendix.
::请参看附录。