PAU Applied Mathematics for Social Sciences II (Cantabria) key terms

Rectangular array of numbers arranged in rows and columns, used to store coefficients or data and to perform operations such as addition, multiplication and inversion.

Scalar value computed from a square matrix that indicates whether the matrix is invertible; it is zero exactly when the matrix is singular.

Matrix that, multiplied by the original square matrix, gives the identity matrix; it exists only when the determinant is nonzero.

Square matrix with ones on the main diagonal and zeros elsewhere; it acts as the neutral element for matrix multiplication.

Maximum number of linearly independent rows or columns; it measures the effective dimension of the information the matrix contains.

Criterion comparing the rank of the coefficient matrix with the rank of the augmented matrix to decide whether a linear system is inconsistent, consistent determined or consistent indeterminate.

Linear system with at least one solution and exactly one; equivalently, the coefficient and augmented matrices share the same rank equal to the number of unknowns.

Linear system with no solution at all; the rank of the augmented matrix exceeds the rank of the coefficient matrix.

Method solving a consistent determined system by writing each unknown as a quotient of determinants, replacing one column by the constants column.

Algorithm that transforms the augmented matrix into row echelon form using elementary row operations, then solves the system by back substitution.

Technique for optimising a linear objective function subject to linear inequality constraints, typically over a polygonal feasible region in two variables.

Linear expression, such as profit or cost, whose maximum or minimum value is sought in an optimisation problem.

Linear inequalities or equations that limit the values the decision variables may take, usually reflecting resource, budget or demand conditions.

Set of points satisfying all constraints simultaneously; in two variables it is usually a convex polygon, possibly unbounded or empty.

Corner point where two boundary lines of the feasible region intersect; the optimum of a linear objective over a bounded region is attained at one of them.

Non-negative variable added to a less-than-or-equal constraint to convert it into an equation, representing unused capacity of a resource.

Value that the outputs of a function approach as the input approaches a given point, regardless of whether the function is defined there.

Limit taken as the input approaches a point from one side only, from the left or from the right; both must agree for the two-sided limit to exist.

Property of a function at a point when its limit there exists, the function is defined there, and both values coincide; otherwise the function has a discontinuity.

Point where the two-sided limit exists but differs from the function value or the function is undefined; the gap can be removed by redefining that single value.

Expression such as zero over zero or infinity minus infinity arising when limits are substituted directly, requiring algebraic manipulation or a standard technique before the limit can be evaluated.

Line that the graph of a function approaches without necessarily touching as the input or the output grows without bound; it may be vertical, horizontal or oblique.

States that a function continuous on a closed interval and taking values of opposite sign at the endpoints vanishes at some interior point.

States that a function continuous on a closed bounded interval attains a maximum and a minimum value somewhere on that interval.

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.

Line touching a curve at a point with the same slope as the derivative there; its equation uses the point and the derivative value.

Rule for differentiating a composite function: the derivative equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Method evaluating indeterminate limits of quotients by replacing the limit of the quotient of functions with the limit of the quotient of their derivatives.

Local maximum or minimum of a function, located where the derivative is zero or undefined and detected by the sign change of the derivative or the second derivative test.

Point where the concavity of a curve changes; for a twice differentiable function it requires the second derivative to be zero and to change sign.

Process of finding the maximum or minimum of a function, often by setting the derivative to zero and checking the nature of the critical points.

Number obtained as the limit of Riemann sums, representing the signed area between a curve and the horizontal axis over an interval.

Family of all antiderivatives of a function, written with an arbitrary additive constant; it is the reverse operation of differentiation.

States that the definite integral of a continuous function over an interval equals the difference between any antiderivative evaluated at the upper and lower endpoints.

Region bounded by two graphs, computed by integrating over the interval the absolute difference of the functions, splitting the interval where they cross.

Probability that an event occurs given that another event has already occurred, computed as the probability of the intersection divided by the probability of the condition.

Situation where the occurrence of one event does not change the probability of another, so the probability of both equals the product of their probabilities.

Rule giving the probability of an event as the sum, over a partition of the sample space, of the conditional probabilities multiplied by the probabilities of the partition pieces.

Formula reversing a conditional probability, expressing the probability of a cause given an observed effect through prior probabilities and likelihoods.

Function assigning a numerical value to each outcome of a random experiment, described by its probability distribution, expectation and variance.

Model for the number of successes in a fixed number of independent trials with constant success probability, characterised by parameters n and p.

Continuous bell-shaped distribution determined by its mean and standard deviation; probabilities are found by standardising to the standard normal and using tables.

Transformation subtracting the mean and dividing by the standard deviation, converting a normal variable into the standard normal so that tables can be used.

Square root of the variance, measuring the average dispersion of the values of a variable around its mean in the original units.