PAU Mathematics II (Galicia) key terms

Matrix obtained by swapping the rows and columns of a given matrix: the entry in row i, column j becomes the entry in row j, column i. Notation A^t.

A square matrix that equals its own transpose, so entry a_ij equals a_ji for every pair of indices. Symmetric matrices have real eigenvalues and are orthogonally diagonalizable.

A scalar assigned to a square matrix that measures whether it is invertible: nonzero means invertible, zero means singular. Computed by cofactor expansion or row reduction.

For a square matrix A with nonzero determinant, the unique matrix A^-1 satisfying A·A^-1 = A^-1·A = I. Computed as the adjugate divided by the determinant.

The transpose of the cofactor matrix of A. Used to compute the inverse as adj(A) divided by det(A) when the determinant is nonzero.

The maximum number of linearly independent rows (equivalently columns) of a matrix. It equals the order of the largest nonzero minor and gives the dimension of the image.

Criterion for classifying a linear system: it is consistent exactly when the rank of the coefficient matrix equals the rank of the augmented matrix; equal to the number of unknowns it has a unique solution, otherwise infinitely many.

A linear system with exactly one solution. This happens when the ranks of the coefficient and augmented matrices coincide and equal the number of unknowns.

A linear system with no solution at all. By Rouché-Frobenius this occurs when the rank of the augmented matrix exceeds the rank of the coefficient matrix.

Method giving each unknown of a Cramer system as a quotient of determinants: the determinant replacing the corresponding column by the constant terms, divided by det(A).

Writing a square matrix A as P·D·P^-1 with D diagonal, possible exactly when A has a full set of linearly independent eigenvectors. The diagonal entries are the eigenvalues.

A scalar λ for which A·v = λ·v has a nonzero vector solution v. Found as roots of the characteristic polynomial det(A − λI) = 0.

A nonzero vector v satisfying A·v = λ·v for some scalar λ. Eigenvectors belonging to distinct eigenvalues are linearly independent.

A set of vectors is linearly independent when no nontrivial linear combination of them equals the zero vector. Equivalently, the matrix formed by them has full column rank.

Operation on two vectors of R³ producing a vector perpendicular to both, with magnitude equal to the area of the parallelogram they span. Components are computed from a formal determinant.

Operation on two vectors giving the real number |u||v|cos θ. It is zero exactly when the vectors are perpendicular, and it yields the angle between them.

The scalar u·(v×w), equal to the signed volume of the parallelepiped spanned by the three vectors. It vanishes exactly when the three vectors are coplanar.

Property of three or more vectors lying in a common plane. For three vectors it is tested by checking that their mixed product is zero.

A nonzero vector perpendicular to a plane, whose components give the coefficients of x, y, z in the general equation of that plane.

Equation of the form ax + by + cz + d = 0, where (a, b, c) is a normal vector to the plane. A point lies on the plane when it satisfies the equation.

The perpendicular distance from a point P to a plane ax+by+cz+d=0, given by |ax₀+by₀+cz₀+d| divided by the square root of a²+b²+c².

Classification of two lines in space as coincident, parallel, intersecting at one point, or skew (crossing without meeting). Determined by comparing direction vectors and solving the system.

A nonzero vector parallel to a line, used to write its parametric and continuous equations. Two lines are parallel exactly when their direction vectors are proportional.

The value a function approaches as x tends to a point from one side only, left or right. A two-sided limit exists exactly when both lateral limits exist and coincide.

A function is continuous at a point when its limit there exists, the function is defined there, and both values agree. On an interval it means continuity at every point.

If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then some c in (a, b) satisfies f(c) = 0. Used to prove existence of roots.

A function continuous on a closed bounded interval attains a maximum and a minimum value there. Guarantees extreme values exist before searching for them.

An expression such as 0/0, ∞/∞, ∞−∞, 0·∞ or 1^∞ arising when substituting into a limit, which requires algebraic manipulation, L'Hôpital's rule, or equivalent infinitesimals to resolve.

For indeterminate quotients 0/0 or ∞/∞, the limit of f/g equals the limit of f'/g' when the latter exists. Requires differentiability near the point.

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 at a point.

The line touching a curve at a point with slope equal to the derivative there. Its equation is y = f(a) + f'(a)(x − a).

Finding maximum or minimum values of a function, typically by solving f'(x) = 0 and classifying the critical points with the sign of the second derivative or a sign table.

If f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) satisfies f'(c) = (f(b) − f(a))/(b − a). Links average and instantaneous rates.

The signed area between a curve and the x-axis over an interval, defined as a limit of Riemann sums. Computed with the fundamental theorem as F(b) − F(a).

A function F whose derivative is the given function f. Any two primitives differ by an additive constant, giving the indefinite integral plus C.

States that if f is continuous on [a, b] and F is any primitive of f, then the definite integral of f from a to b equals F(b) − F(a).

Technique based on the product rule: ∫u dv = uv − ∫v du. Useful for products such as polynomials times exponentials, logarithms, or inverse trigonometric functions.

The region bounded by two functions over an interval, computed as the definite integral of the absolute difference |f(x) − g(x)|, splitting where the curves cross.

Volume generated by rotating a region about an axis, computed by the disk method as π∫[f(x)]²dx, or by washers when a gap exists between curves and axis.

Counting arrangements and selections of objects. Permutations order all elements, variations order a subset, and combinations select a subset without regard to order.

The number of ways to choose k objects from n without order, written C(n,k) and equal to n!/(k!(n−k)!). Rows of Pascal's triangle give these values.

The probability of event A given that B has occurred, P(A|B) = P(A∩B)/P(B) when P(B) > 0. It underlies independence and Bayes' theorem.

Two events are independent when P(A∩B) = P(A)·P(B), equivalently P(A|B) = P(A). Knowing one occurred gives no information about the other.

For a partition of the sample space into events B_i, P(A) equals the sum of P(B_i)·P(A|B_i). Combines conditional probabilities weighted by the partition.

Reverses a conditional probability: P(B_i|A) = P(B_i)·P(A|B_i) divided by the total probability of A. Used to update beliefs after observing A.

Model for the number of successes in n independent 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. Standardizing with z = (x − μ)/σ allows use of the standard normal table.

For large samples, the distribution of the sample mean approaches normality with mean μ and standard error σ/√n, regardless of the population distribution. Justifies normal-based inference.

The standard deviation of a sampling statistic, for the mean equal to σ/√n (or s/√n when σ is unknown). It shrinks as sample size grows.

A range of plausible values for a population parameter, built as estimate ± margin of error. The confidence level is the long-run proportion of intervals containing the parameter.

A decision procedure comparing a null hypothesis H₀ with an alternative H₁ using sample data. The statistic is compared with a critical value or its p-value with the significance level.

The statement of no effect or no difference assumed true while testing, usually written H₀: μ = μ₀. It is rejected only when data are sufficiently unlikely under it.

The probability α, chosen in advance, of rejecting a true null hypothesis. Common values are 0.05 and 0.01; it fixes the critical region of the test.

Rejecting a null hypothesis that is actually true. Its probability is the significance level α, so lowering α reduces this error but makes the test less powerful.