Madrid PAU Applied Mathematics for Social Sciences II flashcards

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

Matrix product condition

The product A·B is defined only when the number of columns of A equals the number of rows of B; the resulting matrix has the rows of A and columns of B.

Invertibility of a square matrix

A square matrix A has an inverse if and only if its determinant is nonzero. Singular matrices (det = 0) have no inverse and correspond to dependent rows.

Rouché–Frobenius theorem

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

Cramer's rule

For a square system with det(A) ≠ 0, each unknown equals the determinant of A with that column replaced by B, divided by det(A).

Homogeneous system nontrivial solutions

A homogeneous system AX = 0 always admits the trivial solution. It admits infinitely many nontrivial solutions exactly when det(A) = 0, i.e. when A is singular.

Corner-point method

For a linear program with a bounded feasible region, the optimum of the objective function is attained at a vertex. Evaluate the objective at every vertex and compare.

Feasible region convexity

The set of points satisfying all linear constraints is the intersection of half-planes, hence convex. Any segment joining two feasible points stays inside the region.

Unbounded linear program

If the feasible region is unbounded and the objective function can increase without limit along a feasible direction, the maximum does not exist and the problem is unbounded.

Integer programming

A linear program where some or all variables must be integers. The optimal integer solution lies inside the feasible region and may differ from the real-valued vertex optimum.

Continuity at a point

A function f is continuous at a when the limit of f as x approaches a exists, equals the function value f(a), and f(a) is defined.

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 limit exists.

Second derivative test

At a critical point where f'(a) = 0, if f''(a) is positive the point is a local minimum; if negative, a local maximum; if zero, the test is inconclusive.

Inflection point condition

A point where the second derivative changes sign marks an inflection point, where concavity switches between upward and downward.

Fundamental theorem of calculus

The definite integral of f from a to b equals F(b) − F(a), where F is any antiderivative of f. It links areas to antiderivatives.

Area between two curves

The area enclosed between y = f(x) and y = g(x) from a to b equals the integral of the absolute difference |f(x) − g(x)| over that interval.

Integration by parts

The formula ∫u dv = uv − ∫v du transforms an integral into a simpler one, useful for products such as x·eˣ or x·ln x.

Conditional probability formula

For events A and B with P(B) > 0, the conditional probability P(A|B) equals P(A ∩ B) divided by P(B).

Total probability theorem

If events A₁, …, Aₙ partition the sample space, then P(B) equals the sum over i of P(Aᵢ)·P(B|Aᵢ).

Bayes' theorem

The posterior probability P(Aᵢ|B) equals P(Aᵢ)·P(B|Aᵢ) divided by the sum of P(Aⱼ)·P(B|Aⱼ) over all partition events.

Independence of two events

Events A and B are independent when P(A ∩ B) = P(A)·P(B), equivalently when P(A|B) = P(A) provided P(B) > 0.

Binomial distribution

For n independent trials each with success probability p, P(X = k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ, with mean np and variance np(1−p).

Standardizing a normal variable

If X is normal with mean μ and standard deviation σ, then z = (x − μ)/σ follows the standard normal, and probabilities are read from Φ(z).

Normal approximation to binomial

When np and n(1−p) are both at least 5, a binomial variable is approximately normal with mean np and standard deviation √(np(1−p)).

Continuity correction

When approximating a discrete binomial by a normal, adjust boundaries by 0.5, so P(X ≤ k) becomes P(X ≤ k + 0.5) before standardizing.

Standard error of the mean

The standard deviation of the sample mean equals σ/√n, where σ is the population standard deviation and n is the sample size.

Confidence interval for a mean

With known σ or large n, the interval is X̄ ± z_(α/2)·σ/√n, where z_(α/2) is 1.96 for 95% confidence.

Confidence interval for a proportion

The interval is p̂ ± z_(α/2)·√(p̂(1−p̂)/n), valid for large samples where the normal approximation to the binomial holds.

Margin of error

The half-width of a confidence interval, E = z_(α/2)·σ/√n. Solving for n gives the sample size needed to achieve a target precision.

Determinant of a 2×2 matrix

For A = [[a, b], [c, d]], the determinant is ad − bc. A is invertible exactly when this value is nonzero.

Rank of a matrix

The rank is the maximum number of linearly independent rows (or columns), equal to the number of nonzero rows in reduced row echelon form.

Objective function in linear programming

The linear expression z = c₁x₁ + … + cₙxₙ to be maximized or minimized, subject to linear constraints and non-negativity conditions.

Central limit theorem

For large sample sizes, the sample mean is approximately normally distributed with mean μ and standard deviation σ/√n, regardless of the population shape.