Madrid PAU Applied Mathematics for Social Sciences II key terms

Matrix obtained by swapping rows and columns of a given matrix: the element in row i, column j moves to row j, column i. Its size is reversed, and transposing twice returns the original matrix.

A single number assigned to a square matrix that measures whether it is invertible. It is zero exactly when the rows or columns are linearly dependent, so the matrix has no inverse.

For a square matrix A with nonzero determinant, the unique matrix A⁻¹ such that A·A⁻¹ = A⁻¹·A = I. Computed as the transposed cofactor matrix divided by the determinant.

The square matrix with ones on the main diagonal and zeros elsewhere. Multiplying any conformable matrix by it leaves that matrix unchanged, so it plays the role of 1 in matrix arithmetic.

Operation combining an m×n matrix with an n×p matrix to give an m×p matrix: each entry is the dot product of a row of the first with a column of the second. It is not commutative.

The number of linearly independent rows, equivalently independent columns, in a matrix. It equals the size of the largest nonzero minor and tells 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: unequal ranks mean no solution, equal ranks below the number of unknowns mean infinitely many.

A 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.

A linear system with infinitely many solutions, described by one or more free parameters. The common rank of coefficient and augmented matrices is smaller than the number of unknowns.

A linear system with no solution at all. The rank of the augmented matrix exceeds the rank of the coefficient matrix, meaning the equations contradict each other.

A square matrix equal to its own transpose, so entry aᵢⱼ equals aⱼᵢ for every pair of indices. Its entries are mirror images across the main diagonal.

An equation whose unknown is a matrix, such as A·X = B. It is solved by isolating X, typically multiplying by an inverse on the correct side and respecting non-commutativity.

The linear expression a linear programming problem seeks to maximize or minimize, usually profit or cost. Its value is evaluated at the vertices of the feasible region to find the optimum.

The set of points satisfying all constraints of a linear programming problem simultaneously. It is a convex polygon, possibly unbounded, and contains every admissible combination of decision variables.

A linear inequality or equation that limits the values the decision variables may take, reflecting resource, budget or demand conditions. All constraints together define the feasible region.

A corner point where two boundary lines of the feasible region intersect. For a linear objective, the optimal value is always attained at one of these vertices, so they are evaluated in turn.

The standard condition that decision variables in a linear programming problem cannot be negative, written x ≥ 0, y ≥ 0. It reflects that quantities such as production or hours cannot be negative.

The point of the feasible region where the objective function reaches its maximum or minimum value. It lies on the boundary, at a vertex or along an entire edge when the objective is parallel to it.

A line along which the objective function keeps a constant value, obtained by setting the objective equal to a parameter. Moving it parallel shows the direction in which the objective grows.

A linear programming problem in which some or all decision variables must take whole-number values, as when counting indivisible items. The optimum may then differ from the continuous relaxation's vertex.

The value a function approaches as the input gets arbitrarily close to a given point, regardless of whether the function is defined there. It exists only when both one-sided limits agree.

The values a function approaches from the left and from the right of a point. A two-sided limit exists exactly when both lateral limits are finite and equal.

A function is continuous at a point when it is defined there, its limit at that point exists, and the limit equals the function value. Graphically, the curve has no break or jump.

The limit of the difference quotient as the increment tends to zero, equal to the slope of the tangent line at that point. It measures the instantaneous rate of change of the function.

The function assigning to each point where it exists the value of the derivative there. It is obtained by applying derivation rules such as product, quotient and chain rules.

The line touching a curve at a point with the same slope as the curve there, given by y = f(a) + f′(a)(x − a). It is the standard linear approximation of the function near that point.

Rule for differentiating a composition: the derivative of f(g(x)) is f′(g(x))·g′(x). It is essential for differentiating powers of expressions, exponentials and logarithms of functions.

Whether a function increases or decreases over an interval, decided by the sign of its derivative: positive means increasing, negative means decreasing. It is analysed by splitting the domain at critical points.

A local maximum or minimum of a function, occurring at a critical point where the derivative is zero or undefined. The sign change of the derivative around the point confirms its type.

A point where the curve changes its concavity, from concave up to concave down or the reverse. It typically satisfies that the second derivative is zero and changes sign there.

A line that the graph of a function approaches without necessarily touching as the input or output grows. Types include vertical, horizontal and oblique asymptotes, found through limits.

If a function is continuous on a closed interval and takes values of opposite sign at the endpoints, then it has at least one root inside. It justifies locating solutions by sign change.

A function continuous on a closed bounded interval attains a maximum and a minimum value somewhere in that interval. It guarantees that global extrema exist under these hypotheses.

The limit of Riemann sums of a function over an interval, interpreted as the signed area between the curve and the horizontal axis. It is computed with the Fundamental Theorem of Calculus.

The definite integral of a continuous function over [a, b] equals F(b) − F(a), where F is any antiderivative. It links area computation to finding primitives.

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

The region enclosed between two graphs from x = a to x = b, computed as the integral of the upper function minus the lower one. Intersection points must be found first to set the limits.

Technique based on the product rule: ∫u dv = uv − ∫v du. It suits products such as a polynomial times an exponential or logarithm, choosing u by the standard priority rule.

The probability that event A occurs given that event B has occurred, written P(A|B) = P(A∩B)/P(B) when P(B) is positive. It restricts the sample space to the outcomes in B.

Two events are independent when the occurrence of one does not change the probability of the other, so P(A∩B) = P(A)·P(B). Equivalently, the conditional probability equals the unconditional one.

Rule giving the probability of an event as a weighted sum over a partition of the sample space: P(A) = Σ P(A|Bᵢ)·P(Bᵢ). It is used when causes or scenarios are mutually exclusive.

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

A random variable taking isolated values, each with an associated probability. Its distribution is described by a probability function summing to one, and by its mean and variance.

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

Continuous symmetric bell-shaped distribution determined by mean μ and standard deviation σ. Probabilities are found by standardizing to the standard normal and reading a table or using technology.

Transforming a normal variable X into the standard normal Z by Z = (X − μ)/σ. This lets any normal probability be read from a single standard normal table.

Approximating B(n, p) by a normal with mean np and standard deviation √(np(1 − p)) when n is large and p is not extreme. A continuity correction of 0.5 is applied.

The probability-weighted average of the values of a random variable, representing its long-run mean. For a discrete variable it is the sum of each value times its probability.

The probability distribution of the sample mean across all possible samples of a fixed size. Its mean equals the population mean and its standard deviation is σ/√n.

For large sample sizes, the distribution of the sample mean is approximately normal regardless of the population's shape. This underpins most confidence intervals and hypothesis tests.

The standard deviation of a sample statistic, for the mean equal to σ/√n or s/√n. It measures how much the statistic varies from sample to sample.

A range built from sample data that contains the unknown population parameter with a stated confidence level. For a mean it is x̄ ± z·σ/√n, with z fixed by the confidence level.

The probability, usually 90%, 95% or 99%, that the procedure producing a confidence interval yields one containing the true parameter. It corresponds to a critical value z of the normal distribution.

A statistic computed from sample data and used to estimate an unknown population parameter, such as the sample mean for the population mean. Good estimators are unbiased and have small variance.

The number of observations drawn, denoted n. It controls the width of confidence intervals, since the error bound decreases in proportion to 1/√n.