Matrices · 矩阵
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| matrix/ˈmeɪtrɪks/ | 矩阵 | jǔ zhèn |
| scalar/ˈskeɪlə/ | 标量 | biāo liàng |
| matrix multiplication/ˈmeɪtrɪks ˌmʌltɪplɪˈkeɪʃn/ | 矩阵乘法 | jǔ zhèn chéng fǎ |
Numbers in a grid
- Sometimes one number is not enough — you have a whole table of them.
- A price list, a set of coordinates, a system of equations: all are grids of numbers.
- A matrix is that grid, treated as a single mathematical object.
- Once packaged this way, whole tables can be added, scaled, and multiplied at once.
网格里的数字
- 有时一个数不够——你有一整张表的数。
- 一张价目表、一组坐标、一个方程组:全都是数字的网格。
- 矩阵就是那个网格,被当作单个数学对象来对待。
- 一旦这样打包,整张表就能被一次性地相加、缩放和相乘。
What a matrix is
- A matrix 矩阵 is a rectangular grid of numbers arranged in rows and columns.
- Its size is written rows × columns: $\begin{bmatrix}1&2\\3&4\end{bmatrix}$ is a $2 \times 2$ matrix.
- Each number inside is an entry, located by its row and column.
- A matrix with one row or one column behaves like a vector.
什么是矩阵
- 矩阵(matrix)是一个由行和列排成的矩形数字网格。
- 它的大小写成 行 × 列:$\begin{bmatrix}1&2\\3&4\end{bmatrix}$ 是一个 $2 \times 2$ 矩阵。
- 里面的每个数是一个元素,由它的行和列定位。
- 只有一行或一列的矩阵,表现得像一个向量。
A matrix is… · 一个矩阵是……
A matrix arranges numbers in a rectangular grid; its size is given as rows × columns. · 一个矩阵按矩形网格排列数字；其大小由行数×列数给出。
A matrix with 2 rows and 3 columns is called a 2 × ____ matrix. · 具有2行和3列的矩阵被称为2 × ____矩阵。
Dimensions are always rows first, then columns: 2 rows, 3 columns is a "2 × 3" matrix. · 维度顺序永远是先行后列：2行3列是“2 × 3”矩阵。
Adding and scaling
- Add two matrices of the same size by adding matching entries.
- $\begin{bmatrix}1&2\\3&4\end{bmatrix} + \begin{bmatrix}5&0\\1&1\end{bmatrix} = \begin{bmatrix}6&2\\4&5\end{bmatrix}$.
- Multiply a matrix by a scalar 标量 by scaling every entry: $2\begin{bmatrix}1&2\\3&4\end{bmatrix} = \begin{bmatrix}2&4\\6&8\end{bmatrix}$.
- These work entry by entry, just like with ordinary numbers.
相加与缩放
- 把两个相同大小的矩阵相加,就把对应位置的元素相加。
- $\begin{bmatrix}1&2\\3&4\end{bmatrix} + \begin{bmatrix}5&0\\1&1\end{bmatrix} = \begin{bmatrix}6&2\\4&5\end{bmatrix}$。
- 把一个矩阵乘以一个标量(scalar),就缩放每一个元素:$2\begin{bmatrix}1&2\\3&4\end{bmatrix} = \begin{bmatrix}2&4\\6&8\end{bmatrix}$。
- 这些都逐个元素地进行,就像对普通的数一样。
To multiply a matrix by a scalar, you… · 要将矩阵乘以标量，你……
Scalar multiplication scales every entry: $2\begin{bmatrix}1&2\\3&4\end{bmatrix} = \begin{bmatrix}2&4\\6&8\end{bmatrix}$. · 标量乘法缩放所有条目：$2\begin{bmatrix}1&2\\3&4\end{bmatrix} = \begin{bmatrix}2&4\\6&8\end{bmatrix}$。
Select all · 所有 true statements about matrices. · 选择关于矩阵的所有正确陈述。
A matrix holds many numbers, not one. The other three are correct. · 矩阵包含多个数字，而非一个。其他三项是正确的。
Matrix multiplication
- Matrix multiplication 矩阵乘法 is different: it combines rows with columns.
- Each entry of the product pairs a row of the first with a column of the second, multiplying and adding.
- The number of columns in the first must equal the number of rows in the second.
- Order matters: $AB$ is usually not the same as $BA$.
矩阵乘法
- 矩阵乘法(matrix multiplication)不一样:它把行与列结合起来。
- 乘积的每个元素把第一个矩阵的一行与第二个矩阵的一列配对,相乘再相加。
- 第一个矩阵的列数必须等于第二个矩阵的行数。
- 顺序重要:$AB$ 通常不等于 $BA$。
A matrix transforms the plane · 矩阵变换平面
A 2x2 matrix moves every point of the plane; this one shears it sideways. · 2x2矩阵移动平面的每一点；这个矩阵使其发生侧向剪切。
Matrix multiplication combines the rows of the first matrix with the columns of the second. · 矩阵乘法将第一个矩阵的行与第二个矩阵的列结合。
Each entry of the product is a row of the first "dotted" with a column of the second — rows-by-columns. · 乘积的每个条目是第一个矩阵的一行“点乘”第二个矩阵的一列——即行乘列。
Matrix multiplication is not entry-by-entry. To find one entry of $AB$, take a whole row of $A$ and a whole column of $B$, multiply matching numbers, and add. Also, the sizes must line up: (columns of $A$) $=$ (rows of $B$), or the product does not exist.
矩阵乘法不是逐个元素进行的。要求出 $AB$ 的一个元素,取 $A$ 的一整行和 $B$ 的一整列,把对应的数相乘,再相加。此外,大小必须对上:($A$ 的列数)$=$($B$ 的行数),否则乘积不存在。
Row-by-column, one entry at a time:
- Top entry: row $(1, 2)$ with column $(5, 6)$ → $5 + 12 = 17$.
- Bottom entry: row $(3, 4)$ with column $(5, 6)$ → $15 + 24 = 39$.
行乘列,一次一个元素:
- 上面的元素:行 $(1, 2)$ 与列 $(5, 6)$ → $5 + 12 = 17$。
- 下面的元素:行 $(3, 4)$ 与列 $(5, 6)$ → $15 + 24 = 39$。
A matrix is a rectangular grid of numbers, sized rows × columns. Add same-size matrices entry by entry; a scalar scales every entry. Matrix multiplication is row-by-column (not entry-by-entry), needs the inner sizes to match, and depends on order.
矩阵是一个矩形的数字网格,大小为 行 × 列。把相同大小的矩阵逐个元素相加;一个标量缩放每一个元素。矩阵乘法是行乘列(不是逐个元素),需要内侧的大小对上,并且依赖顺序。