PAU Andalucía Matemáticas II - fundamentos de práctica key terms

Matrix obtained by turning rows into columns and columns into rows; entry (i,j) of the transpose equals entry (j,i) of the original matrix.

Square matrix that equals its own transpose, so entries mirrored across the main diagonal are identical; A = Aᵗ.

Matrix B such that A·B = B·A = I, the identity. It exists only for square matrices with non-zero determinant, and then B = adj(A)/det(A), with adj(A) defined as the already-transposed cofactor matrix.

Scalar assigned to a square matrix that is zero exactly when the matrix is singular; computed by cofactor expansion or by row operations that track how the value changes.

Determinant of the submatrix left after deleting one row and one column of a square matrix; the associated cofactor multiplies it by (−1)^(i+j).

Transpose of the cofactor matrix of A; dividing it by the determinant of A produces the inverse matrix when that determinant is non-zero.

Maximum number of linearly independent rows, equal to the maximum number of independent columns; found by reducing the matrix to echelon form and counting non-zero rows.

Rule classifying a linear system by comparing the rank of the coefficient matrix with the rank of the augmented matrix and with the number of unknowns.

Linear system with at least one solution and exactly one; occurs when the ranks of coefficient and augmented matrices both equal the number of unknowns.

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

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

Letter appearing in the coefficients or constants of a system; the discussion studies for which values the system is compatible and how many solutions it has.

Non-zero vector parallel to a line; together with one point of the line it determines the line completely in vector, parametric and continuous forms.

Non-zero vector perpendicular to a plane; its components are exactly the coefficients of x, y and z in the general equation of the plane.

Operation on two vectors giving the number |u||v|cos θ; zero result means the vectors are perpendicular, and it yields the angle between them.

Operation on two vectors in space giving a third vector perpendicular to both, with modulus |u||v|sin θ; used to build normals and to compute areas of parallelograms.

Scalar u·(v×w), equal to the determinant of the three vectors as rows; it gives the signed volume of the parallelepiped and vanishes when they are coplanar.

Three or more vectors lying parallel to a single plane; three of them are coplanar exactly when their mixed product is zero.

Form x = a₁ + t·v₁, y = a₂ + t·v₂, z = a₃ + t·v₃, where (a₁,a₂,a₃) is a point, v the direction vector and t the real parameter.

Equation Ax + By + Cz + D = 0, where (A,B,C) is a normal vector; substituting a point of the plane fixes the constant D.

Classification of two lines as coincident, parallel, intersecting or skew, decided by comparing their direction vectors and checking whether a vector joining points lies in their span.

Two lines in space that are neither parallel nor intersecting, so they lie in different parallel planes and share no point.

For a nonzero line direction v and plane normal n, the non-obtuse angle alpha satisfies sin(alpha)=abs(v·n)/(|v||n|). It is complementary to the acute or right angle between the line direction and the normal.

Length of the perpendicular segment from the point to the plane, computed as |Ax₀+By₀+Cz₀+D| divided by the modulus of the normal vector (A,B,C).

Value L that f(x) approaches as x approaches a point; it may exist even when the function is undefined or takes another value at that point.

Property of a function at a point where the limit exists, the function is defined there, and both values coincide; differentiability implies it but not conversely.

Expression such as 0/0, ∞/∞, ∞−∞ or 0·∞ whose limit cannot be read off directly and requires algebraic manipulation, L'Hôpital's rule or equivalent techniques.

Method for indeterminate quotients: replace the limit of f/g by the limit of the quotient of derivatives f'/g', provided the original quotient has form 0/0 or ∞/∞ and the theorem’s differentiability, nonzero denominator-derivative and derivative-limit hypotheses hold.

Limit of the difference quotient, giving the slope of the tangent line and the instantaneous rate of change of a function at a point.

Existence of the derivative at a point; it requires continuity there, and at a breakpoint the two one-sided derivatives must agree.