PAU Applied Math ACS II (La Rioja) key terms

Square matrix in which every entry on the main diagonal is 1 and every other entry is 0. Multiplying any compatible matrix by it leaves that matrix unchanged.

Matrix A⁻¹ that satisfies A·A⁻¹ = A⁻¹·A = I. It exists only for square matrices whose determinant is nonzero.

A scalar computed from a square matrix that indicates whether the matrix is invertible. A zero determinant means the rows are linearly dependent and no inverse exists.

The number of linearly independent rows, equivalently columns, of a matrix. It equals the size of the largest nonzero minor and reveals how many independent equations a system really contains.

Criterion classifying a linear system by comparing the rank of the coefficient matrix with the rank of the augmented matrix and with the number of unknowns.

Linear system with exactly one solution. 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 infinitely many solutions. The ranks of coefficient and augmented matrices agree but are smaller than the number of unknowns, leaving free parameters.

Linear system with no solution at all. The rank of the augmented matrix exceeds the rank of the coefficient matrix, signalling contradictory equations.

Procedure that transforms a system into an equivalent triangular one using elementary row operations, then solves it by back-substitution.

Matrix form reached after Gaussian elimination, in which each leading nonzero entry lies to the right of the leading entry of the row above.

An equation whose unknown is a matrix, typically of the form A·X = B or X·A + B = C. It is solved by isolating X using inverse matrices and respecting non-commutativity.

The set of points satisfying all constraints of a linear programming problem simultaneously. Its shape is a convex polygon, possibly unbounded or empty.

The linear expression, in profit or cost, that the linear programming problem seeks to maximise or minimise over the feasible region.

A linear inequality or equation that limits the values the decision variables may take, usually reflecting resource, demand or non-negativity conditions.

A corner point where two boundary constraint lines intersect. The fundamental theorem of linear programming guarantees that an optimal solution occurs at one of these points.

The unknown quantity, such as number of units produced, that the linear programming model chooses in order to optimise the objective function.

Variant of linear programming in which some or all decision variables must take whole-number values, so the optimum may differ from the continuous relaxation.

The value L that f(x) approaches as x approaches a, regardless of whether f is defined at a. It exists only when both one-sided limits agree.

The value approached by a function when x tends to a point from only the left or only the right. Disagreement between the two signals that the full limit does not exist.

An expression such as 0/0 or ∞−∞ arising from direct substitution that gives no information about the limit and must be resolved by algebra, factorisation or L'Hôpital's rule.

Method for resolving 0/0 or ∞/∞ indeterminate limits by replacing the quotient of functions with the quotient of their derivatives, provided the new limit exists.

Property of a function at a point a requiring that f(a) is defined, the limit as x tends to a exists, and that limit equals f(a).

If a function is continuous on a closed interval and takes values of opposite sign at its endpoints, then it vanishes at some interior point.

A function continuous on a closed interval attains both a maximum and a minimum value somewhere in that interval.

The instantaneous rate of change of a function, defined as the limit of the difference quotient. Geometrically it is the slope of the tangent line.

Line touching a curve at a point with the same slope as the curve there. Its equation uses the derivative evaluated at that point.

A local maximum or minimum of a function. It occurs at critical points where the derivative is zero or undefined, and is classified with the second derivative.

Point where a curve changes from concave to convex or vice versa. It is found where the second derivative changes sign.

Any function F whose derivative equals a given function f. All primitives of f differ only by an additive constant.

The signed area between a curve and the x-axis over an interval, defined as the limit of Riemann sums and computed using primitives.

States that the definite integral of f from a to b equals F(b) − F(a), where F is any primitive of f.

The region enclosed by two functions over an interval, computed as the integral of the absolute difference between them, splitting the interval at their intersection points.

Technique based on the product rule, ∫u dv = uv − ∫v du, used when the integrand is a product of functions such as a polynomial and an exponential.

A function assigning a numerical value to each outcome of a random experiment. It may be discrete or continuous.

Non-negative function whose integral over an interval gives the probability that a continuous random variable falls in that interval. Its total integral equals one.

Continuous symmetric bell-shaped distribution determined by its mean and standard deviation. Probabilities are obtained by standardising to the standard normal variable Z.

Transformation Z = (X − μ)/σ converting a normal variable into the standard normal, allowing probabilities to be read from a single table.

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).

For large samples the distribution of the sample mean approaches a normal distribution regardless of the population's shape, with mean μ and standard error σ/√n.

The standard deviation of a sample statistic, equal to σ/√n for the sample mean. It measures the precision of the estimate and shrinks as sample size grows.

Range of values, built around a sample estimate, that contains the true population parameter with a stated probability such as 95%.

The probability, usually 0.90, 0.95 or 0.99, that the procedure producing a confidence interval yields one containing the true parameter.

The cut-off point of the standard normal or t distribution corresponding to the chosen confidence level, such as 1.96 for 95% in the normal case.

Estimate of the population mean given by the sample mean plus or minus the critical value multiplied by the standard error, using z or t depending on sample size.