PAU Mathematics II (Galicia) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Invertible matrix condition

A square matrix A is invertible exactly when its determinant is nonzero. Otherwise it is singular, its columns are linearly dependent, and the homogeneous system AX = 0 has nontrivial solutions.

Rouché–Capelli consistency criterion

A linear system AX = B is consistent if and only if rank(A) equals rank of the augmented matrix. If that common rank equals the number of unknowns, the solution is unique.

Cramer's rule

For a square system with nonzero determinant, each unknown equals the determinant of the matrix obtained by replacing its column with the constants, divided by det(A).

Determinant of a matrix product

For square matrices of the same order, det(AB) = det(A)·det(B). Consequently det(A^k) = det(A)^k and det(A^{-1}) = 1/det(A) when A is invertible.

Rank via nonzero minors

The rank of a matrix is the order of its largest nonvanishing minor. It also equals the number of nonzero rows after reduction to row echelon form.

Scalar triple product test

Three vectors u, v, w are coplanar exactly when u·(v×w) = 0. The absolute value of this product gives the volume of the parallelepiped they span.

Cross product modulus meaning

The modulus |u×v| equals |u||v|sin θ, the area of the parallelogram spanned by u and v. Half that value is the area of the triangle with those two sides.

Distance from point to plane

For plane Ax+By+Cz+D = 0 and point (x0,y0,z0), the distance is |Ax0+By0+Cz0+D| divided by the square root of A²+B²+C².

Angle between line and plane

If d is the line's direction vector and n the plane's normal, then sin θ = |d·n|/(|d||n|). The angle between two planes uses the angle between their normals.

Skew lines test

Two lines are skew when their direction vectors are not parallel and the scalar triple product of both directions with a vector joining one point on each line is nonzero.

Notable limit sin x over x

As x tends to zero, sin x divided by x tends to 1. Related results: (1−cos x)/x² tends to 1/2 and tan x/x tends to 1.

L'Hôpital's rule

For indeterminate forms 0/0 or ∞/∞, the limit of a quotient equals the limit of the quotient of derivatives, provided that second limit exists.

Differentiability implies continuity

If f is differentiable at a point then it is continuous there. The converse fails: the absolute value function is continuous but not differentiable at zero.

Fermat's necessary extremum condition

At an interior local extremum of a differentiable function, the derivative vanishes. Candidates for extrema are therefore critical points together with endpoints of the domain.

Mean value theorem

If f is continuous on [a,b] and differentiable on (a,b), some interior point c satisfies f'(c) = (f(b)−f(a))/(b−a), so the tangent is parallel to the chord.

Second derivative concavity rule

Where f'' is positive the graph is convex (concave up); where f'' is negative it is concave down. An inflection point requires f'' to change sign.

Fundamental theorem of calculus

If f is continuous on [a,b] and F is any antiderivative of f, the definite integral of f from a to b equals F(b) − F(a).

Area between two curves

The area enclosed by y = f(x) and y = g(x) between their intersection abscissas a and b is the integral from a to b of |f(x) − g(x)|.

Volume of revolution about x-axis

Rotating y = f(x) between a and b about the x-axis gives volume π times the integral from a to b of f(x) squared, the disc method.

Convergence of p-integrals

The improper integral of 1/x^p from 1 to infinity converges exactly when p exceeds 1. On the interval from 0 to 1 it converges exactly when p is below 1.

Integration by parts formula

The integral of u dv equals uv minus the integral of v du. It suits products of polynomials with exponentials, logarithms, or trigonometric functions.

Combinations formula

The number of ways to choose r objects from n distinct objects, order irrelevant, is n! divided by r!(n−r)!. Permutations, where order matters, use n!/(n−r)!.

Independence of two events

Events A and B are independent exactly when P(A∩B) = P(A)P(B), equivalently P(A|B) = P(A) whenever P(B) is nonzero.

Bayes' theorem

For a partition of the sample space, the posterior probability of cause A_i given B equals P(A_i)P(B|A_i) divided by the sum of such products over all causes.

Binomial distribution parameters

For B(n,p), the probability of exactly k successes is C(n,k)p^k(1−p)^{n−k}. The mean is np and the variance is np(1−p).

Complement rule for at least one

The probability that an event occurs at least once equals one minus the probability that it never occurs, which is often far easier to compute directly.

Standard error of the mean

The sample mean has expectation μ and variance σ²/n, so its standard error is σ divided by the square root of n. It decreases as sample size grows.

Confidence interval for a mean

With known σ, the interval is x̄ ± z_{α/2}σ/√n. With unknown σ and small n, replace z by the t critical value with n−1 degrees of freedom and σ by s.

Central limit theorem

For large samples the distribution of the sample mean is approximately normal with mean μ and variance σ²/n, whatever the shape of the population distribution.

Type I and type II errors

A type I error rejects a true null hypothesis and has probability α. A type II error fails to reject a false null hypothesis and has probability β; power is 1 − β.

Sample size for target margin

To achieve margin of error E with known σ, take n equal to (z_{α/2}σ/E)², always rounding upward to the next whole number of observations.

p-value decision rule

The p-value is the probability under the null hypothesis of a result at least as extreme as observed. Reject the null hypothesis when p is less than the significance level α.