Glossary — English / العربية
Every technical word in the course — its Arabic name and what it means in plain language, with flashcards for practice.
Every technical term in the course, with its Arabic name and a plain-language meaning. On every page these words are marked with a dotted underline: hover or tap one to see this information, and press the ع button (bottom-right) to show Arabic names beside the English words throughout the text. The “Used in” links point to the sections where each term appears.
Ch 1 · Systems of Linear EquationsCh 2 · MatricesCh 3 · DeterminantsCh 4 · Vectors in ℝⁿCh 5 · Vector SpacesCh 6 · Linear TransformationsCh 7 · Eigenvalues and EigenvectorsCh 8 · Orthogonality and Least SquaresBasic terms
Ch 1 · Systems of Linear Equations
| English | العربية | Meaning | Used in |
|---|---|---|---|
| augmented matrix | المصفوفة الموسعة | The coefficient matrix of a linear system with the column of constants attached on the right, e.g. $\left[\begin{array}{cc|c}1&2&5\\3&4&6\end{array}\right]$. | 1.1 1.2 1.3 2.1 |
| back-substitution | التعويض العكسي | Solving a triangular system from the last equation upwards, substituting each value found into the equations above it. | 1.1 1.2 2.3 2.4 |
| coefficient matrix | مصفوفة المعاملات | The $m\times n$ matrix $A$ whose entry $a_{ij}$ is the coefficient of $x_j$ in equation $i$ of a linear system. | 1.1 1.2 1.3 2.1 |
| conservation of flow | حفظ التدفق | At every junction of a network, total flow in equals total flow out; each junction gives one linear equation. | 1.3 |
| consistent | متسق | A linear system is consistent if it has at least one solution. | 1.1 1.2 2.4 4.1 |
| elementary row operation | العمليات الصفية الأولية | Interchanging two rows, multiplying a row by a nonzero constant, or adding a multiple of one row to another row; none of them changes the solution set. | 1.1 1.2 2.2 3.2 |
| free variable | متغير حر | A variable whose column contains no pivot. In a consistent system it can take any value (a parameter such as $t$), giving infinitely many solutions. | 1.1 1.2 1.3 2.1 |
| Gaussian elimination | طريقة الحذف لغاوس | Row reduction to a row echelon form (forward phase), followed by back-substitution to solve the system. | 1.1 1.2 2.3 |
| Gauss–Jordan elimination | طريقة غاوس–جوردان للحذف | Row reduction carried all the way to the reduced row echelon form, so that the solution can be read off directly. | 1.2 1.3 3.3 5.4 |
| homogeneous system | نظام متجانس | A linear system $A\mathbf{x}=\mathbf{0}$ in which every constant term is zero. It always has the trivial solution $\mathbf{x}=\mathbf{0}$, and has nontrivial solutions exactly when there is a free variable. | 1.2 1.3 5.1 5.2 |
| inconsistent | غير متسق | A linear system is inconsistent if it has no solution, e.g. elimination produces a false equation $0=c$ with $c\ne0$. | 1.1 1.2 1.3 3.3 |
| interpolating polynomial | كثيرة حدود الاستكمال | The unique polynomial of degree at most $n$ passing through $n+1$ given points with distinct $x$-coordinates. | 1.3 |
| junction | عقدة (نقطة تفرع) | A point of a network where branches meet; also called a node. | 1.3 5.1 5.4 |
| Kirchhoff's laws | قوانين كيرشوف | Current law: at each node, current in equals current out. Voltage law: around each closed loop, battery voltages equal the sum of the drops $IR$. | 1.3 3.3 5.1 5.4 |
| leading variable | متغير رئيسي (أساسي) | A variable whose column in the coefficient matrix is a pivot column; in a consistent system it is determined by the free variables. | 1.2 |
| linear system | نظام معادلات خطية | A finite collection of linear equations $a_1x_1+\cdots+a_nx_n=b$ in the same variables, to be satisfied simultaneously. | 1.1 1.2 1.3 2.1 |
| network | شبكة | A set of junctions (nodes) joined by branches along which something flows — traffic, water, current, money. | 1.3 2.1 2.4 5.1 |
| Ohm's law | قانون أوم | The voltage drop across a resistor is $V=IR$, current times resistance. | 1.3 |
| pivot | المحور (العنصر المحوري) | A position holding a leading entry in an echelon form of a matrix (pivot position); a column containing one is a pivot column. The nonzero number there, used to create zeros, is the pivot. | 1.2 1.3 2.2 2.3 |
| reduced row echelon form | الصيغة الدرجية الصفية المختزلة | A row echelon form in which every leading entry is $1$ and is the only nonzero entry in its column. Every matrix $A$ has exactly one, written $\operatorname{rref}(A)$. | 1.2 2.2 3.2 5.1 |
| row echelon form | الصيغة الدرجية الصفية | A matrix shape in which zero rows are at the bottom and the leading entry of each nonzero row lies strictly to the right of the leading entry of the row above, with zeros below every leading entry (a staircase). | 1.2 2.2 2.3 3.2 |
| row equivalent | متكافئة صفياً | Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. | 1.1 1.2 2.2 3.2 |
| Vandermonde matrix | مصفوفة فاندرموند | The coefficient matrix of an interpolation problem: row $i$ is $1,\ x_i,\ x_i^2,\ \dots,\ x_i^n$. | 1.3 3.2 |
Ch 2 · Matrices
| English | العربية | Meaning | Used in |
|---|---|---|---|
| consumption matrix | مصفوفة الاستهلاك | In the Leontief model, the matrix $C$ whose entry $c_{ij}$ is the input from sector $i$ needed to make one unit of output of sector $j$. | 2.4 |
| elementary matrix | مصفوفة أولية (بسيطة) | A matrix obtained from $I$ by one elementary row operation; multiplying by it on the left performs that row operation. | 2.2 2.3 3.2 |
| final demand | الطلب النهائي | The part of each sector's output wanted by households, government and exports, after the sectors' own use: the vector $\mathbf{d}$ in $(I-C)\mathbf{x}=\mathbf{d}$. | 2.4 |
| flop | عملية بالفاصلة العائمة | One floating-point operation (an addition, subtraction, multiplication or division); used to measure the cost of an algorithm. | 2.3 |
| forward substitution | التعويض الأمامي | Solving a lower triangular system $L\mathbf{y}=\mathbf{b}$ from the first equation down, each equation giving one new unknown. | 2.3 |
| identity matrix | مصفوفة الوحدة (المصفوفة المحايدة) | The square matrix $I_n$ with $1$s on the main diagonal and $0$s elsewhere; $I_mA=A=AI_n$. | 2.1 2.2 2.3 3.3 |
| inverse of a matrix | معكوس المصفوفة | The unique matrix $A^{-1}$ that undoes $A$: $AA^{-1}=A^{-1}A=I$. For $2\times2$ matrices, $A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ when $ad-bc\ne0$. | 5.5 8.3 |
| invertible matrix | مصفوفة قابلة للعكس (غير منفردة) | A square matrix $A$ for which some matrix $A^{-1}$ satisfies $AA^{-1}=A^{-1}A=I$. | 2.1 2.2 2.3 2.4 |
| Invertible Matrix Theorem | مبرهنة المصفوفة القابلة للعكس | For a square matrix $A$: invertible $\iff$ row equivalent to $I$ $\iff$ $A\mathbf{x}=\mathbf{0}$ has only the trivial solution $\iff$ $A\mathbf{x}=\mathbf{b}$ is solvable for every $\mathbf{b}$, and more — all true or all false together. | 2.2 3.2 5.4 5.5 |
| Leontief input–output model | نموذج ليونتيف للمدخلات والمخرجات | The model $\mathbf{x}=C\mathbf{x}+\mathbf{d}$: production equals intermediate demand plus final demand, solved by $\mathbf{x}=(I-C)^{-1}\mathbf{d}$. | 2.4 |
| lower triangular matrix | مصفوفة مثلثية سفلى | A square matrix with only zeros above the main diagonal: $\ell_{ij}=0$ whenever $i\lt j$. | 2.3 3.1 7.1 8.2 |
| LU factorisation | تحليل LU | Writing a square matrix as $A=LU$ with $L$ unit lower triangular (it stores the elimination multipliers) and $U$ upper triangular (the echelon form). | 2.3 |
| main diagonal | القطر الرئيسي | The entries $a_{11},a_{22},\dots,a_{nn}$ of a square matrix, running from top left to bottom right. | 2.1 2.3 3.1 4.2 |
| Markov chain | سلسلة ماركوف | A sequence of state vectors with $\mathbf{x}_{k+1}=P\mathbf{x}_k$ for a fixed stochastic matrix $P$, so $\mathbf{x}_k=P^k\mathbf{x}_0$. | 2.4 7.2 7.3 |
| matrix product | ضرب المصفوفات | For $A$ ($m\times n$) and $B$ ($n\times p$), the $m\times p$ matrix $AB$ with $(AB)_{ij}=\sum_k a_{ik}b_{kj}$ (row $i$ of $A$ times column $j$ of $B$). | 2.1 2.2 3.2 6.1 |
| permutation matrix | مصفوفة التبديل | The identity matrix with its rows rearranged; $PA$ rearranges the rows of $A$ in the same way, and $P^{-1}=P^{T}$. | 2.3 6.3 |
| probability vector | متجه احتمالي | A vector with nonnegative entries that add up to $1$. | 2.4 7.3 |
| regular stochastic matrix | مصفوفة عشوائية منتظمة | A stochastic matrix $P$ some power of which, $P^k$, has all entries strictly positive; its chain converges to a unique steady state. | 2.4 |
| scalar multiple | مضاعف قياسي (ضرب في عدد) | The matrix $cA$ obtained by multiplying every entry of $A$ by the number $c$. | 2.1 3.2 4.1 4.2 |
| singular matrix | مصفوفة منفردة (شاذة) | A square matrix that has no inverse, e.g. one whose reduced echelon form has a row of zeros. | 2.2 2.3 3.2 3.3 |
| square matrix | مصفوفة مربعة | A matrix with the same number of rows and columns ($n\times n$). | 2.1 2.2 2.3 2.4 |
| state vector | متجه الحالة | In a Markov chain, the probability vector $\mathbf{x}_k$ whose $i$-th entry is the probability of being in state $i$ at time $k$. | 2.4 7.3 |
| steady-state vector | متجه الحالة المستقرة | A probability vector $\mathbf{q}$ with $P\mathbf{q}=\mathbf{q}$, found by solving $(P-I)\mathbf{x}=\mathbf{0}$ and scaling so the entries sum to $1$. | 2.4 7.3 |
| stochastic matrix | مصفوفة عشوائية (مصفوفة انتقال) | A square matrix with nonnegative entries whose columns each sum to $1$; entry $p_{ij}$ is the probability of moving from state $j$ to state $i$. | 2.4 7.3 |
| symmetric matrix | مصفوفة متماثلة | A square matrix equal to its own transpose, $A^{T}=A$, i.e. $a_{ij}=a_{ji}$. | 2.1 3.2 3.3 5.1 |
| transpose | منقول المصفوفة | The matrix $A^{T}$ whose rows are the columns of $A$: $\left(A^{T}\right)_{ij}=a_{ji}$. | 2.1 2.2 2.4 3.2 |
| unit lower triangular matrix | مصفوفة مثلثية سفلى أحادية | A lower triangular matrix whose diagonal entries are all $1$; the $L$ in $A=LU$ has this form. | 2.3 |
| upper triangular matrix | مصفوفة مثلثية عليا | A square matrix with only zeros below the main diagonal: $u_{ij}=0$ whenever $i\gt j$. | 2.2 2.3 3.1 3.2 |
| zero matrix | المصفوفة الصفرية | A matrix $O$ whose entries are all $0$; adding it changes nothing: $A+O=A$. | 2.1 5.1 |
Ch 3 · Determinants
| English | العربية | Meaning | Used in |
|---|---|---|---|
| adjugate | المصفوفة المرافقة (المُلحَقة) | The transpose of the cofactor matrix, $\operatorname{adj}A=[C_{ij}]^{T}$. It satisfies $A\,\operatorname{adj}A=(\det A)I$, so $A^{-1}=\operatorname{adj}A/\det A$ when $\det A\ne0$. | 3.3 |
| cofactor | العامل المرافق | The signed minor $C_{ij}=(-1)^{i+j}M_{ij}$; the signs follow a chessboard pattern starting with $+$ in the top-left corner. | 3.1 3.2 3.3 4.3 |
| cofactor expansion | النشر بالعوامل المرافقة (مفكوك لابلاس) | Computing $\det A$ as the sum of the entries of one row (or column) times their cofactors, e.g. $\det A=\sum_j a_{ij}C_{ij}$. Every row and column gives the same answer. | 3.1 3.2 |
| cofactor matrix | مصفوفة العوامل المرافقة | The matrix $[C_{ij}]$ whose $(i,j)$ entry is the cofactor of $a_{ij}$. | 3.3 |
| Cramer's rule | قاعدة كرامر | For an invertible $n\times n$ matrix $A$, the solution of $A\mathbf{x}=\mathbf{b}$ is $x_i=\det A_i(\mathbf{b})/\det A$, where $A_i(\mathbf{b})$ is $A$ with column $i$ replaced by $\mathbf{b}$. | 3.3 |
| determinant | المحدِّدة | A number computed from a square matrix: $ad-bc$ for $\begin{bmatrix}a&b\\c&d\end{bmatrix}$, and by cofactor expansion for larger matrices. It is non-zero exactly when the matrix is invertible. Written $\det A$ or $|A|$. | 2.2 3.1 3.2 3.3 |
| idempotent matrix | مصفوفة جامدة (متساوية القوى) | A square matrix with $A^2=A$, such as a projection. Its determinant is $0$ or $1$. | 3.2 8.3 |
| minor | المحدِّدة الصغرى | The minor $M_{ij}$ of the entry $a_{ij}$ is the determinant of the smaller matrix left after deleting row $i$ and column $j$. | 3.1 3.3 |
| multiplicative property of determinants | خاصية الضرب للمحددات | For square matrices of the same size, $\det(AB)=\det A\,\det B$. It implies $\det(A^{-1})=1/\det A$ and $\det(A^m)=(\det A)^m$. | |
| nilpotent matrix | مصفوفة معدومة القوى | A square matrix with $A^k=O$ for some positive integer $k$. Its determinant is $0$, so it is never invertible. | 3.2 |
| orientation | التوجيه | The sense of turning (anticlockwise or clockwise) of an ordered pair of vectors, or of a figure. A linear map preserves orientation when $\det A\gt 0$ and reverses it, like a mirror, when $\det A\lt 0$. | 3.1 3.2 3.3 4.3 |
| parallelepiped | متوازي السطوح | A solid with six parallelogram faces, spanned by three edge vectors $\mathbf{u},\mathbf{v},\mathbf{w}$ from one corner. Its volume is $\left|\det\begin{bmatrix}\mathbf{u}&\mathbf{v}&\mathbf{w}\end{bmatrix}\right|$. | 3.3 4.3 |
| rule of Sarrus | قاعدة ساروس | A diagonal shortcut for $3\times3$ determinants only: copy the first two columns to the right, add the three down-diagonal products and subtract the three up-diagonal products. | 3.1 |
| skew-symmetric matrix | مصفوفة متخالفة التماثل (شبه متماثلة) | A square matrix with $A^{T}=-A$. Every skew-symmetric matrix of odd size has determinant $0$. | 3.2 6.2 |
| tetrahedron | رباعي السطوح | A solid with four triangular faces. With vertices $\mathbf{0},\mathbf{u},\mathbf{v},\mathbf{w}$ its volume is $\tfrac16\left|\det\begin{bmatrix}\mathbf{u}&\mathbf{v}&\mathbf{w}\end{bmatrix}\right|$. | 3.3 |
| triangular matrix | مصفوفة مثلثية | A square matrix with only zeros below (upper triangular) or above (lower triangular) the main diagonal. Its determinant is the product of its diagonal entries. | 2.3 3.1 3.2 5.1 |
Ch 4 · Vectors in ℝⁿ
| English | العربية | Meaning | Used in |
|---|---|---|---|
| Cauchy–Schwarz inequality | متباينة كوشي–شوارتز | $|\mathbf{u}\cdot\mathbf{v}|\le\|\mathbf{u}\|\|\mathbf{v}\|$ for all $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^n$, with equality exactly when one is a multiple of the other. The same holds with $\langle\mathbf{u},\mathbf{v}\rangle$ in any inner product space. | 4.2 8.1 |
| coplanar | واقعة في مستوى واحد | Lying in one plane. Three vectors are coplanar exactly when their scalar triple product is $0$. | 3.3 4.3 5.2 |
| cosine similarity | تشابه جيب التمام | For nonzero data vectors, $\dfrac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}$ — the cosine of the angle between them, used to compare documents or records regardless of size. | 4.2 |
| cross product | الضرب الاتجاهي | For $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^3$, the vector $\mathbf{u}\times\mathbf{v}=(u_2v_3-u_3v_2,\,u_3v_1-u_1v_3,\,u_1v_2-u_2v_1)$, perpendicular to both, of length equal to the area of the parallelogram they span. | 4.3 6.2 |
| direction vector | متجه الاتجاه | A nonzero vector $\mathbf{d}$ along a line; the line through $P$ is $\mathbf{x}=\mathbf{p}+t\mathbf{d}$. | 4.3 6.4 8.2 |
| dot product | الضرب القياسي | The scalar $\mathbf{u}\cdot\mathbf{v}=u_1v_1+\dots+u_nv_n$; geometrically $\|\mathbf{u}\|\|\mathbf{v}\|\cos\theta$. Also called the scalar product or inner product on $\mathbb{R}^n$. | 2.1 4.2 4.3 5.2 |
| linear combination | تركيب خطي | A vector of the form $c_1\mathbf{v}_1+c_2\mathbf{v}_2+\dots+c_k\mathbf{v}_k$, where $c_1,\dots,c_k$ are scalars. | 2.1 4.1 5.1 5.2 |
| norm | معيار (طول المتجه) | The length of a vector, $\|\mathbf{v}\|=\sqrt{\mathbf{v}\cdot\mathbf{v}}=\sqrt{v_1^2+\dots+v_n^2}$ in $\mathbb{R}^n$; for a general inner product, $\|\mathbf{v}\|=\sqrt{\langle\mathbf{v},\mathbf{v}\rangle}$. | 4.2 8.1 8.2 |
| normal vector | المتجه العمودي (الناظم) | A nonzero vector perpendicular to a plane; for $ax+by+cz=d$ it is $(a,b,c)$. | 4.3 5.3 6.2 8.3 |
| orthogonal | متعامد | Vectors $\mathbf{u},\mathbf{v}$ are orthogonal (perpendicular) when $\mathbf{u}\cdot\mathbf{v}=0$ (for a general inner product, when $\langle\mathbf{u},\mathbf{v}\rangle=0$). | 4.2 4.3 6.3 6.4 |
| orthogonal projection | الإسقاط العمودي | The orthogonal projection of $\mathbf{v}$ onto a nonzero vector $\mathbf{u}$ is $\operatorname{proj}_{\mathbf{u}}\mathbf{v}=\dfrac{\mathbf{v}\cdot\mathbf{u}}{\mathbf{u}\cdot\mathbf{u}}\mathbf{u}$, the vector along $\mathbf{u}$ whose difference from $\mathbf{v}$ is orthogonal to $\mathbf{u}$. Chapter 8 extends this to projection onto a subspace $W$. | 4.2 4.3 6.3 6.4 |
| parallelogram law | قاعدة متوازي الأضلاع | With $\mathbf{u}$ and $\mathbf{v}$ drawn from the same point, $\mathbf{u}+\mathbf{v}$ is the diagonal of the parallelogram they span. | 4.1 |
| position vector | متجه الموضع | The vector $\overrightarrow{OP}$ from the origin to a point $P$; its components are the coordinates of $P$. | 4.1 4.3 |
| right-hand rule | قاعدة اليد اليمنى | Curl the fingers of the right hand from $\mathbf{u}$ towards $\mathbf{v}$; the thumb points along $\mathbf{u}\times\mathbf{v}$. | 4.3 |
| scalar | كمية قياسية | A real number, used to multiply (scale) a vector: $c\mathbf{v}=(cv_1,\dots,cv_n)$. | 2.1 2.2 3.2 4.1 |
| scalar triple product | الضرب الثلاثي القياسي | The number $\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})$, equal to the determinant with rows $\mathbf{u},\mathbf{v},\mathbf{w}$; its absolute value is the volume of the parallelepiped they span. | 4.3 |
| skew lines | مستقيمات متخالفة | Two lines in $\mathbb{R}^3$ that are not parallel and do not intersect. | 4.3 |
| standard unit vector | متجه الوحدة القياسي | One of $\mathbf{e}_1,\dots,\mathbf{e}_n$ in $\mathbb{R}^n$: $\mathbf{e}_i$ has a $1$ in position $i$ and $0$ elsewhere. In $\mathbb{R}^3$: $\mathbf{i},\mathbf{j},\mathbf{k}$. | 4.1 4.3 |
| triangle inequality | متباينة المثلث | $\|\mathbf{u}+\mathbf{v}\|\le\|\mathbf{u}\|+\|\mathbf{v}\|$: one side of a triangle is never longer than the other two together. | 4.2 8.1 |
| unit vector | متجه الوحدة | A vector of norm $1$. Normalising $\mathbf{v}\ne\mathbf{0}$ gives the unit vector $\mathbf{v}/\|\mathbf{v}\|$. | 4.1 4.2 4.3 5.5 |
| vector | متجه | An ordered list of real numbers $(v_1,\dots,v_n)$ in $\mathbb{R}^n$; in the plane or in space it is pictured as an arrow with a length and a direction. From Chapter 5 on, any element of a vector space (a polynomial, a matrix, a function…) is also called a vector. | 1.2 2.1 2.2 2.3 |
| zero vector | المتجه الصفري | The vector $\mathbf{0}$ with $\mathbf{v}+\mathbf{0}=\mathbf{v}$ for every $\mathbf{v}$; in $\mathbb{R}^n$ it is $(0,\dots,0)$, whose components are all zero. Every subspace must contain it. | 2.2 4.1 5.1 6.3 |
Ch 5 · Vector Spaces
| English | العربية | Meaning | Used in |
|---|---|---|---|
| additive inverse | المعكوس الجمعي | For a vector $\mathbf{u}$, the vector $-\mathbf{u}$ with $\mathbf{u}+(-\mathbf{u})=\mathbf{0}$; in every vector space $-\mathbf{u}=(-1)\mathbf{u}$. | 5.1 |
| basis | أساس | A linearly independent set that spans a vector space $V$; every vector of $V$ is then a unique linear combination of the basis vectors. | 5.2 5.3 5.4 5.5 |
| basis theorem | مبرهنة الأساس | In a space of dimension $n$, any $n$ linearly independent vectors form a basis, and any $n$ vectors that span form a basis. | 5.3 |
| change of basis | تغيير الأساس | Converting coordinates from one basis $\mathcal B$ to another $\mathcal C$ with the matrix $P_{\mathcal C\leftarrow\mathcal B}=[\,[\mathbf{b}_1]_{\mathcal C}\ \cdots\ [\mathbf{b}_n]_{\mathcal C}\,]$. | 5.5 6.3 7.2 |
| change-of-coordinates matrix | مصفوفة تغيير الإحداثيات | $P_{\mathcal B}=[\mathbf{b}_1\ \cdots\ \mathbf{b}_n]$, which converts $\mathcal B$-coordinates to standard coordinates: $\mathbf{x}=P_{\mathcal B}[\mathbf{x}]_{\mathcal B}$. | 5.5 7.2 |
| closed under | مغلق تحت (الانغلاق) | A set is closed under an operation if applying the operation to members of the set always gives a member of the set. | 1.3 2.4 5.1 6.2 |
| collinear | على استقامة واحدة | Vectors lying on one line through the origin, i.e. multiples of a single vector; two vectors are dependent exactly when they are collinear. | 4.3 5.2 |
| column space | فضاء الأعمدة | $\operatorname{Col}A$: the span of the columns of $A$, i.e. all vectors $A\mathbf{x}$; a subspace of $\mathbb{R}^m$ for an $m\times n$ matrix. | 5.4 6.2 8.3 |
| coordinate mapping | تطبيق الإحداثيات | The map $\mathbf{x}\mapsto[\mathbf{x}]_{\mathcal B}$ from an $n$-dimensional space $V$ to $\mathbb{R}^n$; it is linear, one-to-one and onto. | 5.5 |
| coordinate vector | متجه الإحداثيات | For a basis $\mathcal B=\{\mathbf{b}_1,\dots,\mathbf{b}_n\}$, the column $[\mathbf{x}]_{\mathcal B}=(c_1,\dots,c_n)$ of the unique weights with $\mathbf{x}=c_1\mathbf{b}_1+\dots+c_n\mathbf{b}_n$. | 5.5 6.2 6.3 |
| dependence relation | علاقة ارتباط خطي | An equation $c_1\mathbf{v}_1+\dots+c_k\mathbf{v}_k=\mathbf{0}$ with weights not all zero. | 5.2 |
| dimension | البُعد | The number of vectors in any basis of a finite-dimensional space $V$, written $\dim V$; for example $\dim\mathbb{R}^n=n$ and $\dim P_n=n+1$. | 4.2 5.1 5.3 5.4 |
| finite-dimensional | منتهي البُعد | A vector space that is spanned by a finite set of vectors, so that it has a basis with finitely many vectors. | 5.3 6.2 6.3 |
| infinite-dimensional | غير منتهي البُعد | A vector space that no finite set spans, such as the space of all polynomials. | 5.3 |
| linearly dependent | مرتبطة خطياً | Vectors for which some weights, not all zero, give $c_1\mathbf{v}_1+\dots+c_k\mathbf{v}_k=\mathbf{0}$; at least one of them is a combination of the others. | 3.2 5.2 5.3 6.1 |
| linearly independent | مستقلة خطياً | Vectors with $c_1\mathbf{v}_1+\dots+c_k\mathbf{v}_k=\mathbf{0}$ only when every $c_i=0$; none of them is a combination of the others. | 1.2 5.2 5.3 5.4 |
| null space | الفضاء الصفري | For an $m\times n$ matrix $A$, $\operatorname{Nul}A=\{\mathbf{x}\in\mathbb{R}^n: A\mathbf{x}=\mathbf{0}\}$, the solution set of the homogeneous system; it is a subspace of $\mathbb{R}^n$. | 5.1 5.4 6.2 6.3 |
| nullity | الصفرية | $\operatorname{nullity}A=\dim\operatorname{Nul}A$, the number of free variables in $A\mathbf{x}=\mathbf{0}$. | 5.4 6.2 6.3 7.1 |
| particular solution | حل خاص | Any one solution $\mathbf{p}$ of $A\mathbf{x}=\mathbf{b}$; every solution is then $\mathbf{p}+\mathbf{v}_h$ with $\mathbf{v}_h\in\operatorname{Nul}A$. | 5.4 |
| rank | الرتبة | $\operatorname{rank}A=\dim\operatorname{Col}A$, the number of pivot positions of $A$; it also equals $\dim\operatorname{Row}A$. | 1.3 2.4 4.2 5.1 |
| Rank–Nullity Theorem | مبرهنة الرتبة والصفرية | For an $m\times n$ matrix, $\operatorname{rank}A+\operatorname{nullity}A=n$. | 5.4 6.2 |
| rotation of axes | دوران المحاور | Describing points in axes turned through an angle $\theta$: $x'=x\cos\theta+y\sin\theta$, $y'=-x\sin\theta+y\cos\theta$. | 5.5 |
| row space | فضاء الصفوف | $\operatorname{Row}A$: the span of the rows of $A$, a subspace of $\mathbb{R}^n$; row operations do not change it. | 5.4 8.1 |
| span | الفضاء المولَّد (المدى الخطي) | $\operatorname{span}\{\mathbf{v}_1,\dots,\mathbf{v}_k\}$ is the set of all linear combinations of the vectors — the smallest subspace containing them. | 2.3 3.1 4.3 5.1 |
| spanning set | مجموعة مولِّدة | A set of vectors whose span is the whole space $V$: every vector of $V$ is a linear combination of them. | 5.1 5.2 5.3 8.1 |
| standard basis | الأساس القياسي | The natural basis of a familiar space: $\mathbf{e}_1,\dots,\mathbf{e}_n$ for $\mathbb{R}^n$, $1,x,\dots,x^n$ for $P_n$, and the matrices $E_{ij}$ for $M_{m\times n}$. | 5.3 5.5 6.1 6.3 |
| subspace | فضاء جزئي | A subset $W$ of a vector space that contains $\mathbf{0}$ and is closed under addition and scalar multiplication, so it is a vector space in its own right. | 5.1 5.3 5.4 6.2 |
| vector space | فضاء متجهات | A nonempty set with an addition and a scalar multiplication that satisfy the ten vector-space axioms; examples are $\mathbb{R}^n$, $P_n$ and $M_{m\times n}$. | 5.1 5.2 5.3 5.5 |
Ch 6 · Linear Transformations
| English | العربية | Meaning | Used in |
|---|---|---|---|
| codomain | المجال المقابل | The space $W$ in which the outputs of a function $T:V\to W$ lie; the range may be smaller than the codomain. | 6.1 6.2 6.3 |
| composition | تركيب الدوال | Applying one map after another: $(S\circ T)(\mathbf{v})=S(T(\mathbf{v}))$. For matrix maps the matrix of $S\circ T$ is $BA$. | 6.1 6.4 |
| dilation | تمدد | Multiplication of every vector by the same factor $k\gt1$ (matrix $kI$); for $0\lt k\lt1$ it is called a contraction. | 6.4 |
| homogeneous coordinates | الإحداثيات المتجانسة | Writing the point $(x,y)$ as $(x,y,1)$ so that translations, as well as linear maps, become multiplication by $3\times3$ matrices. | 6.4 |
| isomorphic | متماثل | Two vector spaces are isomorphic ($V\cong W$) if there is an isomorphism between them; finite-dimensional spaces are isomorphic exactly when they have the same dimension. | 6.2 |
| isomorphism | تماثل (تشاكل تقابلي) | A linear map that is both one-to-one and onto; it has a linear inverse. | 5.5 6.2 |
| kernel | النواة | The set $\ker T=\{\mathbf{v}:T(\mathbf{v})=\mathbf{0}\}$ of vectors a linear map sends to zero; a subspace of the domain. For $T(\mathbf{x})=A\mathbf{x}$ it is $\operatorname{Nul}A$. | 6.2 6.3 |
| linear operator | مؤثر خطي | A linear transformation from a vector space to itself, $T:V\to V$. | 6.1 6.3 |
| linear transformation | تحويل خطي | A function $T:V\to W$ between vector spaces with $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ and $T(c\mathbf{u})=cT(\mathbf{u})$ for all vectors and scalars. | 3.3 5.5 6.1 6.2 |
| matrix of a linear transformation | مصفوفة التحويل الخطي | For $T:V\to W$ and bases $\mathcal B$, $\mathcal C$, the matrix $[T]_{\mathcal C\leftarrow\mathcal B}$ whose $j$-th column is $[T(\mathbf{b}_j)]_{\mathcal C}$; it satisfies $[T(\mathbf{x})]_{\mathcal C}=[T]_{\mathcal C\leftarrow\mathcal B}[\mathbf{x}]_{\mathcal B}$. | 6.1 7.2 |
| matrix transformation | تحويل مصفوفي | The map $\mathbf{x}\mapsto A\mathbf{x}$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ given by an $m\times n$ matrix $A$; it is always linear. | 6.1 6.2 |
| one-to-one | متباين (أحادي) | A map with $T(\mathbf{u})=T(\mathbf{v})$ only when $\mathbf{u}=\mathbf{v}$. A linear map is one-to-one exactly when $\ker T=\{\mathbf{0}\}$. | 5.5 6.2 |
| onto | شامل | A map $T:V\to W$ whose range is all of $W$: every vector of $W$ is an output. | 5.5 6.2 |
| principal axes | المحاور الرئيسية | A basis of directions in which the matrix of a (usually symmetric) operator, such as a stress matrix, becomes diagonal. | 6.3 8.4 |
| range of a linear transformation | مدى التحويل الخطي | The set $\operatorname{range}T=\{T(\mathbf{v}):\mathbf{v}\in V\}$ of all outputs; a subspace of the codomain. For $T(\mathbf{x})=A\mathbf{x}$ it is $\operatorname{Col}A$. | 6.2 |
| reflection | انعكاس | The map that sends each point to its mirror image in a line; in a line through the origin at angle $\theta$ its matrix is $\begin{bmatrix}\cos2\theta&\sin2\theta\\\sin2\theta&-\cos2\theta\end{bmatrix}$, with determinant $-1$. | 3.2 3.3 6.1 6.3 |
| rotation matrix | مصفوفة الدوران | The matrix $R_\theta=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$ that turns every vector of the plane anticlockwise through the angle $\theta$ about the origin. | 2.2 3.1 5.5 6.4 |
| shear | قص | A map that slides points parallel to an axis by an amount proportional to their distance from it, e.g. $(x,y)\mapsto(x+ky,\,y)$; it keeps area (determinant $1$). | 2.2 3.1 3.2 3.3 |
| similar matrices | مصفوفات متشابهة | Square matrices $A$ and $B$ with $B=P^{-1}AP$ for some invertible $P$; they represent the same linear operator in two different bases and share determinant, trace and rank. | 6.3 7.2 |
| similarity transformation | تحويل التشابه | Replacing a square matrix $A$ by $P^{-1}AP$ with $P$ invertible — the effect on a matrix of changing the basis. | 6.3 |
| standard matrix | المصفوفة القياسية | For a linear $T:\mathbb{R}^n\to\mathbb{R}^m$, the matrix $A=[T(\mathbf{e}_1)\ \cdots\ T(\mathbf{e}_n)]$ with $T(\mathbf{x})=A\mathbf{x}$ for all $\mathbf{x}$. | 6.1 6.2 6.3 6.4 |
| translation | انسحاب | The map $\mathbf{x}\mapsto\mathbf{x}+\mathbf{h}$ that moves every point by the same vector $\mathbf{h}$; it is not linear when $\mathbf{h}\ne\mathbf{0}$. | 6.1 6.4 |
Ch 7 · Eigenvalues and Eigenvectors
| English | العربية | Meaning | Used in |
|---|---|---|---|
| algebraic multiplicity | التعددية الجبرية | The number of times an eigenvalue occurs as a root of the characteristic polynomial. | 7.1 7.2 |
| attractor | جاذب | For $\mathbf{x}_{k+1}=A\mathbf{x}_k$, the origin is an attractor when every eigenvalue has $|\lambda|\lt1$: all trajectories tend to $\mathbf{0}$. | 7.3 |
| characteristic equation | المعادلة المميزة | The equation $\det(A-\lambda I)=0$, whose solutions are exactly the eigenvalues of $A$. | 7.1 |
| characteristic polynomial | كثيرة الحدود المميزة | The polynomial $p(\lambda)=\det(A-\lambda I)$ of degree $n$; its roots are the eigenvalues of $A$. | 7.1 7.2 7.3 8.4 |
| defective matrix | مصفوفة معيبة (غير قابلة للتقطير) | A square matrix with fewer than $n$ independent eigenvectors, so it cannot be diagonalised, e.g. $\begin{bmatrix}1&1\\0&1\end{bmatrix}$. | 7.2 |
| diagonal matrix | مصفوفة قطرية | A square matrix whose entries off the main diagonal are all $0$; its powers are found by raising each diagonal entry to the power. | 2.2 3.1 3.3 6.3 |
| diagonalisable | قابلة للتقطير | A square matrix $A$ is diagonalisable if $A=PDP^{-1}$ for an invertible $P$ and a diagonal $D$ — equivalently, if it has $n$ linearly independent eigenvectors. | 7.2 7.3 8.4 |
| diagonalisation | التقطير (تحويل المصفوفة إلى الشكل القطري) | Finding $P$ (eigenvectors as columns) and $D$ (matching eigenvalues on the diagonal) with $A=PDP^{-1}$. | 6.3 7.1 7.2 8.4 |
| dominant eigenvalue | القيمة الذاتية المهيمنة | An eigenvalue whose absolute value is strictly larger than that of every other eigenvalue; it sets the long-run growth rate of $A^k\mathbf{x}$. | 7.3 |
| dynamical system | نظام ديناميكي (متقطع) | A rule that updates a state vector step by step, here $\mathbf{x}_{k+1}=A\mathbf{x}_k$, so that $\mathbf{x}_k=A^k\mathbf{x}_0$. | 7.3 |
| eigenspace | الفضاء الذاتي | For an eigenvalue $\lambda$, the subspace $E_\lambda=\operatorname{Nul}(A-\lambda I)$: all eigenvectors for $\lambda$ together with $\mathbf{0}$. | 7.1 7.2 8.3 8.4 |
| eigenvalue | القيمة الذاتية | A scalar $\lambda$ for which $A\mathbf{x}=\lambda\mathbf{x}$ has a nonzero solution $\mathbf{x}$: the factor by which $A$ stretches (or flips) a special direction. | 2.4 7.1 7.2 7.3 |
| eigenvector | المتجه الذاتي | A nonzero vector $\mathbf{x}$ with $A\mathbf{x}=\lambda\mathbf{x}$ for some scalar $\lambda$; the matrix does not change its direction (only its length, or its sense when $\lambda\lt0$). | 6.3 7.1 7.2 7.3 |
| eigenvector basis | أساس من المتجهات الذاتية | A basis of $\mathbb{R}^n$ made of eigenvectors of $A$; in this basis the map $\mathbf{x}\mapsto A\mathbf{x}$ has a diagonal matrix. | 7.2 |
| geometric multiplicity | التعددية الهندسية | The dimension of the eigenspace of an eigenvalue, i.e. the number of independent eigenvectors it has; always between $1$ and the algebraic multiplicity. | 7.1 7.2 |
| Leslie model | نموذج ليزلي | An age-structured population model $\mathbf{x}_{k+1}=L\mathbf{x}_k$: the first row of $L$ holds birth rates, the subdiagonal holds survival rates; its dominant eigenvalue is the long-run growth factor. | 7.3 |
| power method | طريقة القوى | An iterative method that multiplies a vector by $A$ again and again (rescaling each time) to approximate the dominant eigenvalue and its eigenvector. | 7.3 |
| repeller | طارد (مُنفِّر) | The origin is a repeller when every eigenvalue has $|\lambda|\gt1$: all nonzero trajectories move away from $\mathbf{0}$. | 7.3 |
| saddle point | نقطة سرجية | The origin is a saddle point when one eigenvalue has $|\lambda|\gt1$ and another $|\lambda|\lt1$: trajectories come in along one eigenvector and leave along another. | 7.3 |
| trajectory | مسار | The sequence of states $\mathbf{x}_0,\mathbf{x}_1,\mathbf{x}_2,\dots$ produced by a dynamical system from one starting point. | 7.3 |
Ch 8 · Orthogonality and Least Squares
| English | العربية | Meaning | Used in |
|---|---|---|---|
| best approximation | أفضل تقريب | The orthogonal projection $\hat{\mathbf{y}}=\operatorname{proj}_W\mathbf{y}$ is the point of $W$ closest to $\mathbf{y}$; $\|\mathbf{y}-\hat{\mathbf{y}}\|$ is the distance from $\mathbf{y}$ to $W$. | 8.3 |
| design matrix | مصفوفة التصميم | In data fitting, the matrix whose $i$-th row holds the basis functions evaluated at $x_i$, e.g. $\begin{bmatrix}1&x_i\end{bmatrix}$ for a line. | 8.3 |
| Gram–Schmidt process | عملية غرام-شميت | An algorithm that turns a basis $\mathbf{x}_1,\dots,\mathbf{x}_p$ into an orthogonal basis of the same subspace by subtracting from each $\mathbf{x}_k$ its projections onto the vectors already found. | 8.2 8.3 8.4 |
| inner product | الضرب الداخلي | A rule $\langle\mathbf{u},\mathbf{v}\rangle$ giving a number for each pair of vectors that is symmetric, linear in each slot and positive ($\langle\mathbf{u},\mathbf{u}\rangle\gt0$ for $\mathbf{u}\ne\mathbf{0}$); the dot product is the standard example. | 8.1 8.2 |
| inner product space | فضاء الضرب الداخلي | A vector space together with an inner product, so that lengths, distances and angles make sense in it. | 8.1 8.2 |
| least-squares line | خط المربعات الصغرى | The line $y=\hat c_0+\hat c_1x$ that minimises the sum of the squared vertical distances (residuals) from the data points. | 8.3 |
| least-squares solution | حل المربعات الصغرى | A vector $\hat{\mathbf{x}}$ that makes $\|\mathbf{b}-A\mathbf{x}\|$ as small as possible; it solves the normal equations $A^{T}A\hat{\mathbf{x}}=A^{T}\mathbf{b}$. | 8.3 |
| Legendre polynomials | كثيرات حدود لوجاندر | The orthogonal polynomials $1,\ t,\ t^2-\tfrac13,\ t^3-\tfrac35t,\dots$ (up to scaling) obtained by Gram–Schmidt from $1,t,t^2,\dots$ with $\langle f,g\rangle=\int_{-1}^1fg\,dt$. | 8.2 |
| normal equations | المعادلات الطبيعية | The square system $A^{T}A\hat{\mathbf{x}}=A^{T}\mathbf{b}$, whose solutions are exactly the least-squares solutions of $A\mathbf{x}=\mathbf{b}$. | 4.3 8.3 |
| orthogonal basis | أساس متعامد | A basis that is also an orthogonal set; coordinates in it are $c_j=\dfrac{\mathbf{y}\cdot\mathbf{u}_j}{\mathbf{u}_j\cdot\mathbf{u}_j}$. | 6.3 8.1 8.2 8.3 |
| orthogonal complement | المتممة العمودية | For a subspace $W$ of $\mathbb{R}^n$, the subspace $W^{\perp}$ of all vectors orthogonal to every vector in $W$; for example $(\operatorname{Row}A)^{\perp}=\operatorname{Nul}A$. | 8.1 8.2 8.3 |
| orthogonal decomposition | التحليل المتعامد | Writing a vector as $\mathbf{y}=\hat{\mathbf{y}}+\mathbf{z}$ with $\hat{\mathbf{y}}$ in a subspace $W$ and $\mathbf{z}$ in its orthogonal complement $W^{\perp}$; the decomposition is unique. | 8.3 |
| orthogonal matrix | مصفوفة متعامدة | A square matrix $U$ whose columns are orthonormal, so $U^{T}U=I$ and $U^{-1}=U^{T}$; it preserves lengths and angles. | 8.1 8.2 8.4 |
| orthogonal set | مجموعة متعامدة | A set of vectors in which every two different vectors are orthogonal: $\mathbf{u}_i\cdot\mathbf{u}_j=0$ for $i\ne j$. | 8.1 8.2 |
| orthogonally diagonalisable | قابلة للتقطير المتعامد | A matrix that can be written $A=PDP^{T}$ with $P$ orthogonal and $D$ diagonal; for real matrices this happens exactly when $A$ is symmetric. | 8.4 |
| orthonormal basis | أساس متعامد معياري | A basis whose vectors are mutually orthogonal unit vectors; coordinates in it are simply $c_j=\mathbf{y}\cdot\mathbf{q}_j$. | 8.1 8.2 8.3 8.4 |
| orthonormal set | مجموعة متعامدة معيارية | An orthogonal set in which every vector has length $1$. | 8.1 |
| positive definite | موجبة التحديد | A quadratic form (or symmetric matrix) with $\mathbf{x}^{T}A\mathbf{x}\gt0$ for all $\mathbf{x}\ne\mathbf{0}$; equivalently all eigenvalues are positive. | 8.4 |
| projection matrix | مصفوفة الإسقاط | The matrix $P=A\left(A^{T}A\right)^{-1}A^{T}$ that sends every vector to its orthogonal projection onto $\operatorname{Col}A$ (columns of $A$ independent); it satisfies $P^{T}=P$ and $P^2=P$. | 8.3 8.4 |
| QR factorisation | تحليل QR | Writing a matrix with independent columns as $A=QR$, where $Q$ has orthonormal columns spanning $\operatorname{Col}A$ and $R=Q^{T}A$ is upper triangular and invertible. | 8.2 8.3 |
| quadratic form | الصيغة التربيعية | A function $Q(\mathbf{x})=\mathbf{x}^{T}A\mathbf{x}$ with $A$ symmetric; every term has degree two, e.g. $6x_1^2-4x_1x_2+9x_2^2$. | 8.4 |
| residual | البواقي | The difference between an observed value and the value predicted by a model; the residual vector is $\mathbf{b}-A\hat{\mathbf{x}}$. | 8.3 |
| singular value | القيمة المفردة | For any matrix $A$, the numbers $\sigma_i=\sqrt{\lambda_i}$ where $\lambda_i$ are the eigenvalues of $A^{T}A$; they appear in the singular value decomposition $A=U\Sigma V^{T}$. | 8.4 |
| spectral decomposition | التحليل الطيفي | Writing a symmetric matrix as $A=\sum\lambda_i\mathbf{u}_i\mathbf{u}_i^{T}$, a sum of eigenvalues times projections onto orthonormal eigenvectors. | 8.4 |
| Spectral Theorem | مبرهنة الطيف | Every real symmetric matrix has real eigenvalues, mutually orthogonal eigenspaces, and an orthogonal diagonalisation $A=PDP^{T}$. | 6.3 8.4 |
Basic terms
| English | العربية | Meaning | Used in |
|---|---|---|---|
| absolute value | القيمة المطلقة | The distance of a number from $0$; it is never negative. $|-3|=3$. | 2.3 3.3 4.3 5.5 |
| closed interval | فترة مغلقة | An interval that INCLUDES its end points, written $[a,b]$. | |
| coefficient | معامل | The number multiplying a variable, e.g. $7$ in $7x^2$. | 1.1 1.2 1.3 2.1 |
| conjugate | المرافق | For $\sqrt{a}-b$, the conjugate is $\sqrt{a}+b$; multiplying them removes the square root. | |
| constant | ثابت | A fixed number that does not change, e.g. $5$, $\pi$ or $C$. | 1.1 1.2 1.3 4.2 |
| degree | درجة | For a polynomial: the highest power of $x$. For angles: a unit of measure ($360^\circ$ in a circle). | 1.3 4.2 5.1 5.3 |
| domain | مجال الدالة | All the $x$-values you are allowed to put into a function. | 6.1 6.2 6.3 |
| equation | معادلة | A statement that two expressions are equal, e.g. $2x+1=7$. | 1.1 1.2 1.3 2.1 |
| exponential function | دالة أُسّية | A function with the variable in the power, e.g. $e^x$ or $2^x$. | |
| factor | التحليل إلى عوامل | Writing an expression as a product, e.g. $x^2-4=(x-2)(x+2)$. | 7.1 7.3 |
| function | دالة | A rule that gives exactly one output $y$ for each input $x$. Think of a machine: put $x$ in, get $f(x)$ out. | 1.1 1.3 3.1 4.2 |
| graph | التمثيل البياني | The picture of all points $(x, f(x))$ of a function drawn on axes. | 1.3 4.1 5.4 6.3 |
| identity | متطابقة | An equation true for every allowed value, e.g. $\sin^2 x+\cos^2 x=1$. | 1.1 2.1 2.2 2.3 |
| interval | فترة | All the numbers between two end points, e.g. $[1,3]$ or $(0,\infty)$. | 5.1 8.1 |
| inverse function | دالة عكسية | A function that undoes another: if $f(2)=5$ then $f^{-1}(5)=2$. | |
| inverse trigonometric | مثلثية عكسية | $\sin^{-1}, \cos^{-1}, \tan^{-1}, \ldots$ — they return the angle whose sine/cosine/tangent is given. | |
| linear function | دالة خطية | A function whose graph is a straight line: $f(x)=mx+b$. | |
| logarithm | لوغاريتم | The inverse of an exponential: $\log_b y = x$ means $b^x = y$. | 8.3 |
| natural logarithm | اللوغاريتم الطبيعي | The logarithm with base $e\approx 2.718$, written $\ln x$. | |
| open interval | فترة مفتوحة | An interval that does NOT include its end points, written $(a,b)$. | |
| parabola | قطع مكافئ | The U-shaped graph of a quadratic function. | 1.3 2.3 3.2 5.1 |
| parameter | وسيط (بارامتر) | An extra variable (often $t$) that both $x$ and $y$ depend on. | 1.1 1.2 3.3 5.3 |
| piecewise | دالة متعددة القواعد | A function given by different formulas on different parts of its domain. | |
| polynomial | كثيرة الحدود | A sum of terms like $3x^2-5x+1$ using only whole-number powers of $x$. | 1.3 5.1 5.2 5.3 |
| quadratic | تربيعية | Of degree 2, e.g. $x^2-3x+2$; its graph is a parabola. | 4.2 6.3 8.4 |
| radian | راديان | Angle unit used in calculus: $2\pi$ radians $=360^\circ$. | 4.2 |
| range | مدى الدالة | All the $y$-values (outputs) a function actually produces. | 1.1 1.3 3.2 6.2 |
| rational function | دالة نسبية | A fraction whose top and bottom are polynomials, e.g. $\dfrac{x+1}{x-5}$. | 1.3 |
| root | جذر / صفر | A value of $x$ that makes $f(x)=0$ — where the graph meets the $x$-axis. | 2.3 6.1 7.1 7.2 |
| theorem | مبرهنة (نظرية) | A mathematical statement that has been proved to be always true. | 1.1 1.2 1.3 2.1 |
| trigonometric function | دالة مثلثية | $\sin, \cos, \tan, \cot, \sec, \csc$ — functions built from angles in a right triangle / unit circle. | |
| unit circle | دائرة الوحدة | The circle of radius $1$ centred at the origin; the point at angle $\theta$ is $(\cos\theta, \sin\theta)$. | 8.4 |
| variable | متغيّر | A letter such as $x$ or $t$ that stands for a quantity that can change. | 1.1 1.2 1.3 2.1 |