8.7 Systems of Equations and Matrices: Glossary

Costas Loizou; Shuwei Chen; Deak Kirkham

8.7 Systems of Equations and Matrices: Glossary

The pronunciation guide given in these glossaries draws on the symbols International Phonetic Alphabet. We have made two adaptations to the IPA system. Firstly, we use slanted lines as opposed to square brackets to enclose pronunciation material. We do this as the slanted line convention is more common in language teaching contexts. Secondly, we use the symbol <r> to denote the voiced alveolar approximant sound which appears in British English at the beginning of the each word in the phrase 'red rhombus'; the IPA would instead use an inverted 'r' symbols namely <ɹ>. We have done this to avoid unnecessary complication to the visuals of the glossaries.

In addition to these conventions, please also note that the pronunciations given are indicative of contemporary Standard British English pronunciation and should not be read as normative, nor as a target pronunciation that you must aim to produce. The pronunciation of English varies widely across World Englishes; part of becoming fluent in the academic English of mathematics is developing an awareness of, and sensitivity to, the variation in pronunciation that you will encounter.

Word	Pronunciation	Part of speech词性	Related terminology	Chinese meaning	Example sentence 1	Example sentence 2
matrix (plural: matrices)	/ˈmeɪtrɪks/	n.	复数 matrices； transpose matrix 转置矩阵; identity matrix 单位矩阵； augmented matrix 增广矩阵； square matrix 方阵； inverse matrix 逆矩阵； diagonal matrix 对角矩阵； two-by-two matrix 二阶矩阵	矩阵	The augmented matrix of \(A x = b\) is \([A \mid b]\). 方程组 \(A x = b\) 的增广矩阵为 \([A \mid b]\)。	We multiply a matrix \(A\) by scalar \(k\) to get \(kA\). 我们用数 \(k\) 乘矩阵 \(A\) 得到 \(kA\)。
array	/əˈreɪ/	n.		数组	A matrix is a rectangular array of numbers arranged in rows and columns. 矩阵是由行和列排列的矩形数字数组。	We store the components of a vector in a one-dimensional array. 我们将向量的分量存储在一维数组中。
entry	/ˈɛntri/	n.		矩阵元素	The \((i,j)\)-th entry of matrix \(A\) is denoted \(a_{ij}\). 矩阵 \(A\) 的第 \(i\) 行第 \(j\) 列元素记为 \(a_{ij}\)。	All diagonal entries of \(I\) equal \(1\). 单位矩阵 \(I\) 所有对角元素均为 \(1\)。
row	/rəʊ/	n.		行	The first row of the matrix is \((1,2,3)\). 该矩阵的第一行是 \((1,2,3)\)。	We add corresponding entries in each row. 我们在每一行中对应元素相加。
column	/ˈkɒləm/	n.		列	Each column can be treated as a vector. 每一列都可以看作一个向量。	In a matrix, each entry is located by its row and column index. 在矩阵中，每个元素由行号和列号确定位置。
pivot	/ˈpɪvət/	n./v.	pivot column 主元列； pivot variable 主变量	主元	The column containing a pivot is a pivot column. 包含主元的列叫做主元列。	Each pivot position has zeros above and below in RREF. 简化行阶梯形中每个主元上下均为 \(0\)。
free variable	/friː ˈveəriəbəl/	n. phr.	dependent variable 非自由变量	自由变量	Let \(x_3=t\) be a free variable for parametric solution. 令 \(x_3=t\) 作为自由变量写出参数解。	More free variables lead to more solution parameters. 自由变量越多，解的参数越多。
determinant	/dɪˈtɜːmɪnənt/	n.		行列式	Row swap changes the sign of the determinant. 交换行会改变行列式的符号。	The determinant of a diagonal matrix is the product of diagonal entries. 对角矩阵的行列式等于对角元素之积。
singular	/ˈsɪŋɡjʊlə/	adj.	non-singular 非奇异的	奇异的	A square matrix is singular if \(det(A)=0\). 方阵满足 \(det(A)=0\) 时为奇异矩阵。	The system \(A x = 0\) has nontrivial solutions if \(A\) is singular. 若 \(A\) 奇异，则 \(A x = 0\) 有非零解。
invertible	/ɪnˈvɜːtɪbəl/	adj.	invertible matrix 可逆矩阵； non-invertible 不可逆的	可逆的	A square matrix \(A\) is invertible if there exists \(A^{-1}\) such that \(A A^{-1}=A^{-1} A=I\). 若存在 \(A^{-1}\) 满足 \(AA^{-1}=A^{-1}A=I\)，则方阵 \(A\) 可逆。	An invertible matrix has linearly independent columns. 可逆矩阵的列向量线性无关。
row operation	/rəʊ ˌɒpəˈreɪʃən/	n. phr.	elementary row operations 初等行运算； row reduction 行化简	行运算	Elementary row operations include row swap, row scaling and row addition. 初等行变换包括换行、数乘行、行倍加。	Use row operations to reduce \(A\) to RREF. 通过行变换将矩阵 \(A\) 化为简化行阶梯形。
Gaussian elimination	/ˈɡaʊsiən ɪˌlɪmɪˈneɪʃən/	n. phr.		高斯消元法	Gaussian elimination solves \(A x = b\) via elementary row operations. 高斯消元借助初等行变换求解方程组 \(A x = b\)。	We apply Gaussian elimination to an \(n \times n\) square matrix. 我们对 \(n \times n\) 方阵执行高斯消元。
reduced row echelon form	/rɪˈdjuːst rəʊ ˈekəlɒn fɔːm/	n. phr.	简写RREF； row echelon form 行阶梯形	行最简形矩阵	Convert matrix \(A\) into reduced row echelon form (RREF) by row operations. 用行变换将矩阵 \(A\) 化为简化行阶梯形（RREF）。	In reduced row echelon form, each pivot entry is \(1\). 简化行阶梯形中每个主元元素都为 \(1\)。
system of equations	/ˈsɪstəm əv ɪˈkweɪʒənz/	n. phr.	linear system of equations 线性方程组	方程组	Write the system of equations in matrix form \(A x = b\). 将方程组写成矩阵形式 \(A x = b\)。	This system of equations has infinitely many solutions. 该方程组有无穷多组解。
consistent system	/kənˈsɪstənt ˈsɪstəm/	n. phr.	consistent adj. 一致的，相容的； inconsistent system 不相容方程组	相容方程组	A consistent system has either one solution or infinitely many solutions. 相容方程组有唯一解或无穷多解。	Use RREF to check if the system is consistent. 用简化行阶梯形判断方程组是否相容。
general solution	/ˈdʒɛnərəl səˈluːʃən/	n. phr.	specific/particular solutions opposite	通解	Write the general solution of \(A x = 0\) in vector form. 把 \(A x = 0\) 的通解写成向量形式。	Read the general solution directly from RREF. 可从简化行阶梯形直接写出通解。
particular solution	/pəˈtɪkjələ səˈluːʃn/	n. phr.		特解	We add the particular solution to the general solution. 我们将特解加到通解中。	A particular solution satisfies the non-homogeneous equation. 特解满足非齐次方程。
root	/ruːt/	n.		根，解	Every polynomial equation has at least one root in complex numbers. 每个多项式方程在复数中至少有一个根。	The root of \(x^2-4=0\) is \(x=2\) or \(x=-2\). 方程\(x^2-4=0\)的根是\(2\)或\(-2\)。
left/right-hand side	/left raɪt hænd saɪd/	n. phr.	LHS/RHS	左/右端	We simplify both left-hand side and right-hand side separately. 我们分别化简左右两边。	The left-hand side equals the right-hand side. 左边等于右边。
unknown	/ʌnˈnəʊn/	n./adj.	variables 未知数	未知量	We solve for the unknown using algebraic methods. 我们用代数方法求解未知量。	Let \(x\) be an unknown in the equation \(x+3=7\). 设\(x\)为方程\(x+3=7\)中的未知数。
coefficient	/ˌkəʊɪˈfɪʃənt/	n.		系数	The coefficient of \(x\) in \(3x+2\) is \(3\). \(3x+2\)中\(x\)的系数是\(3\)。	A coefficient determines the rate of change of the term. 系数决定该项的变化率。
constant	/ˈkɒnstənt/	n.	constant of integration 积分常数	常数	The constant term does not depend on \(x\). 常数项不依赖于\(x\)。	The speed is constant throughout the motion. 运动过程中速度保持不变。
manipulation	/məˌnɪpjuˈleɪʃn/	n.		运算，处理	Algebraic manipulation simplifies the expression. 代数变形可简化表达式。	Careful manipulation of terms leads to the solution. 对项的处理得到解。
expansion	/ɪkˈspænʃn/	n.		展开	The expansion of \((x+1)^2\) is \(x^2+2x+1\). \((x+1)^2\)展开为\(x^2+2x+1\)。	Binomial expansion is used for higher powers. 二项式展开用于高次幂。
linear transformation	/ˈlɪniə trænsfɔːˈmeɪʃən/	n. phr.		线性变换	Reflection is a type of linear transformation in \(\mathbb{R}^2\). 反射是二维空间中的一种线性变换。	A linear transformation maps the zero vector to zero vector. 线性变换把零向量映射为零向量。
transpose	/trænsˈpəʊz/	v.	n. /ˈtrænspəʊz/	转置；转置矩阵	We take the transpose to swap rows and columns. 取转置可以交换矩阵的行与列。	\((AB)^T=B^T A^T\) holds for matrix transpose. 矩阵转置满足 \((AB)^T=B^T A^T\)。
right inverse	/raɪt ˈɪnvɜːs/	n. phr.	left inverse 左逆映射 / 左逆矩阵	右逆映射，右逆矩阵	If \(AB=I\), then \(B\) is a right inverse of \(A\). 若 \(AB=I\)，则称 \(B\) 是 \(A\) 的右逆。	An invertible matrix has identical left and right inverse. 可逆矩阵的左逆与右逆相等。
commutative	/kəˈmjuːtətɪv/	adj.	non-commutative 非交换的； anti-commutative 反交换的	可交换的	Matrix addition is commutative: \(A+B=B+A\). 矩阵加法可交换：\(A+B=B+A\)。	Diagonal matrices are commutative under multiplication. 对角矩阵乘法运算可交换。
associative	/əˈsəʊʃiətɪv/	adj.	associative property 结合律	结合的	Vector addition is associative: \(({\bf u}+{\bf v})+{\bf w}={\bf u}+({\bf v}+{\bf w})\). 向量加法满足结合律：\(({\bf u}+{\bf v})+{\bf w}={\bf u}+({\bf v}+{\bf w})\)。	Matrix multiplication is associative: \((AB)C=A(BC)\). 矩阵乘法满足结合律：\((AB)C=A(BC)\)。
distributive	/dɪˈstrɪbjʊtɪv/	adj.	distributive property 分配律	分配的	Scalar multiplication is distributive over vector addition: \(k({\bf u}+{\bf v})=k{\bf u}+k{\bf v}\). 数乘对向量加法满足分配律：\(k({\bf u}+{\bf v})=k{\bf u}+k{\bf v}\)。	Expansion \((a+b)(c+d)\) follows distributive law. 展开式 \((a+b)(c+d)\) 遵循分配律。
main diagonal	/meɪn daɪˈæɡənəl/	n. phr.		主对角线	A diagonal matrix has nonzeros only on the main diagonal. 对角矩阵仅在主对角线上有非零元素。	Reflect over the main diagonal to get matrix transpose. 沿主对角线对称即为矩阵转置。
symmetric	/sɪˈmɛtrɪk/	adj.	non-symmetric 非对称的； anti-symmetric 反对称的；	对称的	Matrix \(A\) is symmetric if \(A^T=A\). 满足 \(A^T=A\) 的矩阵 \(A\) 是对称矩阵。	The graph of \(y=x^2\) is symmetric about the \(y\)-axis. 曲线 \(y=x^2\) 关于 \(y\) 轴对称。
formula	/ˈfɔːmjələ/	n.		公式	The quadratic formula solves \(ax^2+bx+c=0\). 二次公式用于解方程\(ax^2+bx+c=0\)。	We apply a formula to compute the area. 我们用公式计算面积。
recursion	/rɪˈkɜːʃn/	n.		递归	The sequence is defined by recursion. 该数列通过递推定义。	Recursion expresses each term in terms of previous ones. 递归用前一项定义后一项。
linear	/ˈlɪniə/	adj.	linear function 线性函数； non-linear 非线性的； linear mapping 线性映射	线性的	We solve a linear system \(A x = b\). 我们求解线性方程组 \(A x = b\)。	A linear transformation satisfies \(T(k{\bf u}+{\bf v})=kT({\bf u})+T({\bf v})\). 线性变换满足 \(T(k{\bf u}+{\bf v})=kT({\bf u})+T({\bf v})\)。
binary	/ˈbaɪnəri/	adj.		二元的	Dot product is a binary operation on vector space. 点积是向量空间上的二元运算。	A binary matrix has entries only \(0\) and \(1\). 二元矩阵的元素只有 \(0\) 和 \(1\)。
algebra	/ˈældʒɪbrə/	n.		代数	Algebra involves manipulating symbols and equations. 代数涉及符号与方程的运算。	Algebra provides methods for solving unknown variables. 代数用于求解未知量。