PAU Applied Math ACS II (Aragón) 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.

Number of rows and columns of a matrix, written as m×n; it fixes whether two matrices can be added or multiplied together.

Matrix obtained by exchanging rows and columns of a given matrix; the element in row i, column j moves to row j, column i.

Square matrix that equals its own transpose, so the entry in row i, column j is identical to the entry in row j, column i.

Scalar associated with a square matrix that indicates whether it is invertible and gives the area or volume scaling factor of the associated linear transformation.

For a square matrix A with nonzero determinant, the matrix A⁻¹ satisfying A·A⁻¹ = A⁻¹·A = I; it allows solving matrix equations by multiplication.

Square matrix with ones on the main diagonal and zeros elsewhere; multiplying any compatible matrix by it leaves that matrix unchanged.

Operation combining an m×n matrix with an n×p matrix to give an m×p matrix, where each entry is the sum of products of a row and a column.

Maximum number of linearly independent rows (or columns) of a matrix; it reveals how many independent equations a system really contains.

Criterion classifying a linear system as inconsistent, determined or indeterminate by comparing the rank of the coefficient matrix with the rank of the augmented matrix.

Linear system with exactly one solution, occurring when the ranks of coefficient and augmented matrices coincide and equal the number of unknowns.

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

Linear system with no solution, detected when the rank of the augmented matrix exceeds the rank of the coefficient matrix.

Elimination procedure that transforms a system into an equivalent triangular form by combining rows, then solves it by back-substitution.

Set of points satisfying all constraints of a linear programming problem simultaneously; it is a convex polygon or polyhedron when constraints are linear.

Linear expression whose value must be maximised or minimised over the feasible region in a linear programming problem, typically profit, cost or time.

Linear inequality or equation that limits the values the decision variables may take, usually reflecting limited resources, demand or non-negativity conditions.

Quantity the modeller chooses in an optimisation problem, such as number of units produced, and which the objective function and constraints are written in terms of.

Corner point where two boundary lines of the feasible region intersect; an optimal solution of a linear program always occurs at one of these points.

Line on which the objective function keeps a constant value; sliding it parallel across the feasible region identifies the optimal vertex.

Value L that f(x) approaches as x approaches a, regardless of whether the function is defined at that point; written lim f(x) = L.

Limit taken approaching the point from one side only, left or right; a two-sided limit exists exactly when both lateral limits exist and are equal.

Expression such as 0/0 or ∞−∞ arising from direct substitution that gives no information about the limit and must be resolved by algebraic manipulation or a rule.

Method for indeterminate quotients 0/0 or ∞/∞ that replaces the limit of the quotient by the limit of the quotient of the derivatives, when that limit exists.

Property of a function at a point where the limit exists, the function is defined there, and both values coincide; failure of any condition gives a discontinuity.

Point where the two-sided limit exists but differs from the function value or the function is undefined; redefining a single value would make it continuous.

Instantaneous rate of change of a function, defined as the limit of the difference quotient; geometrically it gives the slope of the tangent line.

Line touching a curve at a point with slope equal to the derivative there; its equation is y = f(a) + f′(a)(x − a).

Rule for differentiating a composite function: the derivative of f(g(x)) is f′(g(x)) multiplied by g′(x).

Interior point of the domain where the derivative is zero or does not exist; candidates for local maxima, minima or inflection points with horizontal tangent.

Process of finding maximum or minimum values of a function, typically by locating critical points, checking the sign of the derivative and comparing endpoint values.

Function F whose derivative equals a given function f; every continuous function has infinitely many primitives differing by an additive constant.

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.

Fundamental result stating that the definite integral of a continuous function equals the difference between any primitive evaluated at the upper and lower limits.

Region bounded by two graphs, computed by integrating the absolute difference of the functions between their intersection points.

Probability that event A occurs given that event B has occurred, defined as P(A∩B)/P(B) whenever P(B) is not zero.

Formula reversing conditional probabilities, giving P(A|B) from the prior probability of A and the likelihoods of B under each possible cause.

Rule computing the probability of an event by summing its conditional probabilities over a partition of the sample space, weighting each by the probability of its conditioning event.

Two events whose joint probability equals the product of their individual probabilities, so knowing one occurred does not change the probability of the other.

Model for the number of successes in n independent identical trials with constant success probability p; its mean is np and 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 using tables.

Transformation z = (x − μ)/σ converting a normal variable into the standard normal with mean zero and standard deviation one, so tables can be used.

Use of a normal distribution with mean np and standard deviation √(np(1−p)) to estimate binomial probabilities when n is large and p is not extreme.

Range of plausible values for an unknown population parameter, built from a sample statistic and a margin of error at a chosen confidence level.

Probability, usually 90%, 95% or 99%, that the procedure producing a confidence interval yields an interval containing the true parameter.

Standard deviation of a sample statistic such as the mean, equal to σ/√n when the population standard deviation is known; it measures sampling variability.

Result stating that sample means from a large sample are approximately normally distributed around the population mean, whatever the shape of the population distribution.

Procedure comparing a null hypothesis about a parameter with sample data, deciding whether to reject it using a test statistic and a significance level.

Starting assumption in a hypothesis test, usually that a parameter equals a stated value or that two parameters are equal; it is rejected or not rejected from sample evidence.

Maximum probability, usually 0.05 or 0.01, of rejecting a true null hypothesis that the researcher is willing to accept in a hypothesis test.