PAU Applied Math ACS II (Galicia) key terms

Square matrix whose entries are 1 on the main diagonal and 0 everywhere else. Multiplying any conformable matrix by it leaves that matrix unchanged, so it plays the role of the number 1 in matrix algebra.

Matrix A⁻¹ that satisfies A·A⁻¹ = A⁻¹·A = I. It exists only for square matrices whose determinant is non-zero, and it allows a matrix equation to be solved by multiplying on the left.

Number assigned to a square matrix by a fixed rule (Sarrus for order 3, cofactor expansion in general). It is non-zero exactly when the matrix is invertible and the associated homogeneous system has only the trivial solution.

Maximum number of linearly independent rows (equivalently columns) of a matrix. It equals the number of non-zero rows in reduced row echelon form and decides whether a linear system is consistent.

Criterion classifying a linear system by comparing the rank of the coefficient matrix with the rank of the augmented matrix: unequal ranks mean no solution, equal ranks with fewer than unknowns mean infinitely many, equal to the number of unknowns means a unique solution.

Linear system with exactly one solution. By Rouché-Frobenius this happens when the rank of the coefficient matrix equals the rank of the augmented matrix and both equal the number of unknowns.

Linear system with no solution at all. Geometrically the equations describe parallel lines, parallel planes or figures with no common point, and algebraically the ranks of coefficient and augmented matrices differ.

Linear system in which every constant term is zero. It is always consistent because the zero vector solves it; it has non-trivial solutions only when the determinant of the coefficient matrix vanishes.

Method giving each unknown of a compatible determined system as a quotient of determinants: the determinant of the coefficient matrix replaces one column with the constants, divided by the coefficient determinant.

Procedure that reduces the augmented matrix of a system to reduced row echelon form using elementary row operations, after which the solution (or the absence of one) can be read directly.

Value L that the outputs of f approach as the input approaches a, regardless of whether f is defined at a. It exists only when the left-hand and right-hand limits agree.

Property that f is defined at a, the limit of f as x tends to a exists, and that limit equals f(a). If any of the three conditions fails, the function has a removable, jump or infinite discontinuity there.

Expression such as 0/0 or ∞/∞ obtained by direct substitution into a limit, which gives no information. It must be resolved by factoring, rationalising or applying L'Hôpital's rule.

Technique for indeterminate quotients 0/0 or ∞/∞ stating that the limit of f/g equals the limit of the quotient of derivatives f'/g', provided the latter limit exists.

Limit of the difference quotient as the increment tends to zero. It measures the instantaneous rate of change of the function and equals the slope of the tangent line at that point.

Function that assigns to each x the value f'(x) of the derivative, obtained from the original function using the rules for sums, products, quotients and composition.

Rule for differentiating a composite function: the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). It is essential for powers of expressions and for exponential and logarithmic compositions.

Line touching a curve at a point with the same slope as the curve there. Its equation is y = f(a) + f'(a)(x − a), and it gives the best linear approximation near that point.

Line perpendicular to the tangent at a point of a curve. Its slope is the negative reciprocal of f'(a), so it is vertical whenever the tangent is horizontal.

Process of finding the maximum or minimum of a function, usually by solving f'(x) = 0 and classifying the critical points with the sign change of the derivative or the second derivative test.

Point where the curve changes from concave to convex or the reverse. At such a point the second derivative is zero or undefined and changes sign.

Vertical line x = a approached by the graph of a function whose values tend to plus or minus infinity as x tends to a. It typically appears where a denominator vanishes.

If f is continuous on a closed interval and takes values of opposite sign at its endpoints, then some interior point of the interval satisfies f(c) = 0. It justifies the existence of roots.

Function F whose derivative is the given function f. Any two primitives of the same function differ only by an additive constant, which is why the indefinite integral carries a constant of integration.

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

Fundamental theorem statement used to evaluate a definite integral: if F is any primitive of f, the integral from a to b equals F(b) − F(a).

Region bounded by two functions on an interval, computed as the integral of the upper function minus the lower one. Intersection points must be found first to fix the limits of integration.

Probability that event A occurs given that B has occurred, written P(A|B) and defined as P(A ∩ B)/P(B) whenever P(B) is positive. It restricts the sample space to B.

Situation in which knowing one event occurred does not change the probability of the other, formally P(A ∩ B) = P(A)·P(B), equivalent to P(A|B) = P(A).

Rule giving the probability of an event as a weighted sum over a partition of the sample space: P(A) = Σ P(Bᵢ)·P(A|Bᵢ). It is the denominator when applying Bayes' theorem.

Formula reversing a conditional probability: P(Bᵢ|A) = P(Bᵢ)·P(A|Bᵢ) divided by the total probability of A. It updates prior beliefs after observing evidence.

Model for the number of successes in n independent repetitions of a Bernoulli trial 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, symmetric about μ. Probabilities are found by standardising and reading the standard normal table.

Change of variable z = (x − μ)/σ that turns any normal variable into the standard normal with mean 0 and standard deviation 1, allowing table values to be used.

Use of a normal distribution with the same mean and variance as a binomial when n is large and p is not extreme, applying a continuity correction of half a unit to the endpoints.

Sum of all observations divided by their number. It is the most common measure of central tendency but is strongly affected by extreme values.

Square root of the variance, expressing dispersion in the same units as the data. It measures the typical distance of observations from the mean.

Ratio of the standard deviation to the mean, often given as a percentage. It compares dispersion across data sets with different units or very different means.

Number between −1 and 1 measuring the strength and direction of the linear relation between two variables. Values near zero indicate weak linear association.

Average of the products of paired deviations from their respective means. Its sign shows the direction of the linear relation, but its size depends on the units of the variables.

Least-squares line y = a + bx predicting one variable from the other, with slope b = covarianza/varianza de x. It passes through the point of the means.

Set of points satisfying all the constraints of a linear programming problem simultaneously. When non-empty and bounded it is a convex polygon whose vertices are candidate optimal solutions.

Linear expression in the decision variables that is to be maximised or minimised subject to the constraints. Its level curves are parallel lines whose direction shows how the value changes.

Inequality or equation that limits the values of the decision variables, usually coming from resource limits or minimum requirements. Each one defines a half-plane whose intersection forms the feasible region.

Corner point where two boundary lines of the feasible region meet. If an optimum exists, at least one vertex attains it, so evaluating the objective there is enough.

Simplified representation of a real social or economic situation using variables, equations and inequalities, built so that conclusions drawn from it can be interpreted in the original context.