PAU Applied Mathematics for Social Sciences II - Canary Islands / ULPGC & ULL key terms

Rectangular array of real numbers arranged in rows and columns. Its size is given as rows by columns, and it is used to store coefficients and data compactly.

Maximum number of linearly independent rows, equal to the maximum number of linearly independent columns. It measures how much genuine information the matrix carries and decides whether a linear system has solutions.

Number assigned to a square matrix, computed by cofactor expansion or by row reduction. It is nonzero exactly when the matrix is invertible, so it detects unique solutions of square systems.

The matrix A⁻¹ that satisfies A·A⁻¹ = A⁻¹·A = I. It exists only for square matrices with nonzero determinant and is found by the adjugate method or by Gauss–Jordan elimination.

Systematic method that uses elementary row operations to transform an augmented matrix into row-echelon form, after which the unknowns are obtained by back substitution.

Matrix formed by joining the coefficient matrix of a system with the column of constant terms. Comparing its rank with the coefficient matrix's rank classifies the system as incompatible, compatible determined, or compatible indeterminate.

Criterion stating that a linear system is consistent exactly when the rank of the coefficient matrix equals the rank of the augmented matrix. If both ranks equal the number of unknowns, the solution is unique.

Method giving each unknown of a square compatible system as a quotient of two determinants: the determinant with that column replaced by the constants, divided by the coefficient determinant.

Linear system having exactly one solution. For n unknowns this happens when the coefficient and augmented matrices share rank n.

Linear system with infinitely many solutions, arising when the common rank of coefficient and augmented matrices is smaller than the number of unknowns. Solutions are written with free parameters.

Linear system with no solution at all. It occurs when the rank of the augmented matrix is strictly greater than the rank of the coefficient matrix.

An equation in which the unknown is a matrix, such as A·X + B = C. It is solved by isolating X using inverse matrices and the algebraic rules of matrix products.

The linear expression, in the decision variables, that a linear programming problem seeks to maximise or minimise, for instance profit or cost.

Set of points satisfying all the constraints of a linear programming problem simultaneously. It is a convex polygon, possibly unbounded or empty, in two-variable problems.

Linear inequality or equation that limits the values the decision variables may take, usually expressing resource availability, minimum demand, or non-negativity of quantities.

Corner point where two boundary lines of the feasible region intersect. For a linear objective, the optimum over a bounded region is always attained at one of these points.

Technique for two-variable linear programming that draws the constraints to obtain the feasible region, then evaluates the objective at each vertex to identify the optimum.

Line along which the objective function keeps a constant value. Sliding it parallel across the feasible region shows where the maximum or minimum is reached.

Optimisation model in which a linear objective is maximised or minimised subject to linear constraints, typically representing production planning, transport, or diet problems.

Difference between the amount of a resource available and the amount actually used by an optimal solution. Zero slack marks a binding constraint whose relaxation could improve the objective.

Study of how the optimal solution and optimal value change when coefficients of the objective or of the constraints are modified within certain ranges.

Value L that a function's outputs approach as the input approaches a point or grows without bound. It describes local and asymptotic behaviour even where the function is undefined.

Expression such as 0/0, ∞/∞, or ∞ − ∞ obtained by direct substitution, which gives no information about the limit and must be resolved by factoring, conjugates, or L'Hôpital's rule.

Result stating that a 0/0 or ∞/∞ indeterminate limit of a quotient equals the limit of the quotient of the derivatives, provided that the latter exists.

Line that a curve approaches arbitrarily closely as the input tends to a finite value or to infinity. Vertical, horizontal, and oblique types are distinguished by the corresponding limits.

Property of a function at a point where the limit exists, the function is defined there, and both values coincide. Polynomial, rational, exponential, and logarithmic functions are continuous on their domains.

Instantaneous rate of change of a function at a point, defined as the limit of the difference quotient. Geometrically it is the slope of the tangent line to the graph.

Line touching a curve at a point with the same slope as the derivative there. Its equation is y = f(a) + f′(a)(x − a).

Measure of how much a function changes per unit of input. The average rate uses a secant line over an interval; the instantaneous rate is the derivative at a point.

Behaviour of a function as increasing or decreasing on an interval. It is decided by the sign of the first derivative: positive means increasing, negative means decreasing.

Local maximum or minimum of a function. Candidates are critical points where the derivative is zero or undefined, confirmed by sign changes of the derivative or the second-derivative test.

Curvature property of a graph determined by the sign of the second derivative: positive gives convex (cup-shaped) arcs, negative gives concave (cap-shaped) arcs.

Point where the graph changes from concave to convex or the reverse. It requires a zero or undefined second derivative and a genuine change of sign there.

Process of finding the maximum or minimum of a modelled function, for example cost, revenue, or profit, by locating critical points and comparing values on the relevant interval.

Function whose derivative equals a given function. Any two primitives of the same function differ only by an additive constant, giving the indefinite integral.

Number obtained as the limit of Riemann sums of a function over an interval. It represents the signed area between the graph and the horizontal axis.

Formula evaluating a definite integral as F(b) − F(a), where F is any primitive of the integrand. It links areas under curves with antiderivatives.

Region enclosed between two graphs over an interval, computed as the definite integral of the upper function minus the lower one, splitting the interval where they cross.

Function giving the probability that a random variable takes a value less than or equal to x. It increases from 0 to 1 and characterises the distribution completely.

Numerical function defined on a sample space that assigns a real value to each outcome. It may be discrete, taking isolated values, or continuous, taking values in an interval.

Probability-weighted average of the values of a random variable, interpreted as the long-run mean. For discrete variables it is the sum of value times probability.

Expected value of the squared deviation of a random variable from its mean. It measures dispersion, and its square root is the standard deviation.

Square root of the variance, expressing dispersion in the same units as the variable. It is the key scale factor when standardising a normal variable.

Model for the number of successes in n independent trials with constant success probability p. Its mean is np and its variance np(1 − p).

Continuous bell-shaped distribution determined by mean μ and standard deviation σ. Probabilities are found by standardising to the standard normal and reading a table.

Change of variable z = (x − μ)/σ that converts a normal variable into the standard normal with mean 0 and standard deviation 1, allowing use of a single probability table.

Subset of the sample space, that is, a collection of possible outcomes of a random experiment. Events are combined with union, intersection, and complement.

Set of all possible outcomes of a random experiment. Its elements are the elementary events, and any event is a subset of it.

Rule assigning to an event the quotient of favourable equally likely outcomes over total equally likely outcomes. It applies only when all elementary outcomes have the same probability.

Counting techniques, including permutations, variations, and combinations, used to count outcomes or favourable cases when the sample space is too large to list.

Probability of an event A given that event B has occurred, defined as P(A∩B)/P(B) when P(B) is positive. It updates information about the experiment.

Situation in which the occurrence of one event does not alter the probability of another, so P(A∩B) = P(A)·P(B).

Rule computing the probability of an event by summing its conditional probabilities over a partition of the sample space: P(A) = Σ P(Bᵢ)·P(A|Bᵢ).

Formula that reverses a conditional probability, giving P(Bᵢ|A) from prior probabilities P(Bᵢ) and likelihoods P(A|Bᵢ). It is used to revise beliefs after observing evidence.

Result stating that a function continuous on a closed interval and taking values of opposite sign at its endpoints must vanish somewhere inside. It justifies existence of roots.