Kaoyan Mathematics II - independent mathematical foundations key terms

A set closed under addition and scalar multiplication, satisfying axioms such as associativity, commutativity, and existence of a zero vector and additive inverses.

A property of a set of vectors where no nontrivial linear combination equals the zero vector. Equivalently, no vector in the set is a combination of the others.

A linearly independent set of vectors that spans a vector space, so every element is expressed uniquely as a linear combination of them. The number of basis vectors is the dimension.

The dimension of the column space of a matrix, equal to the number of linearly independent rows or columns. It determines whether a linear system has unique, many, or no solutions.

A scalar computed from a square matrix that is nonzero exactly when the matrix is invertible. It also gives the signed volume scaling factor of the associated linear map.

A scalar for which some nonzero vector is mapped to a scalar multiple of itself by a linear operator. Eigenvalues characterize stability, vibration modes, and diagonalizability.

Writing a matrix as a product involving a diagonal matrix and a matrix of eigenvectors. It is possible exactly when there are enough linearly independent eigenvectors.

A scalar operation on two vectors equal to the sum of products of corresponding components. It relates to the cosine of the angle between them and tests orthogonality.

A vector operation defined in three dimensions, producing a vector perpendicular to both inputs with magnitude equal to the area of the spanned parallelogram.

The component of one vector along the direction of another, computed using the dot product. It underlies least squares fitting and Gram-Schmidt orthogonalization.

The value a function approaches as its input approaches a chosen point, whether or not the function is defined there. It is the foundation for continuity, derivatives, and integrals.

A property of a function whose value at a point equals the limit of the function there. Informally, the graph has no break, jump, or hole at that point.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of a difference quotient. Geometrically it gives the slope of the tangent line.

The property that a function has a well-defined derivative at a point. Differentiability at a point implies continuity there, but continuity does not imply differentiability.

An interior point of a function's domain where the derivative is zero or fails to exist. Such points are candidates for local maxima, minima, or inflection behavior.

The way a curve bends, determined by the sign of the second derivative. Positive second derivative means concave up; negative means concave down; a sign change marks an inflection point.

The signed area between a function's graph and the horizontal axis over an interval, defined as a limit of Riemann sums. It yields a number, unlike the indefinite integral.

The result linking differentiation and integration: differentiating an accumulated area function recovers the integrand, and a definite integral equals the change in an antiderivative.

An integral over an unbounded interval or of a function with an infinite discontinuity. It is defined by a limit and either converges to a finite value or diverges.

A power series formally built from the derivatives of a function at a point. A smooth function need not equal its Taylor series; Taylor polynomial error estimates require their stated differentiability and remainder hypotheses.

The property of a sequence or series whose terms approach a finite limit or whose partial sums approach a finite sum. Divergence means no such finite limit exists.

The derivative of a multivariable function with respect to one variable while holding the others fixed. It measures sensitivity to that single input direction.

The vector of all first partial derivatives of a scalar function. Where the gradient is nonzero and the function is differentiable, its direction gives steepest increase in the Euclidean norm and is normal to the regular level set.

The matrix of first partial derivatives of a vector-valued function, describing its best linear approximation. Its determinant measures local volume scaling under the transformation.

An equation relating an unknown function of one variable to its derivatives. Its order is the highest derivative present; solutions form families determined by initial conditions.

A differential equation together with specified values of the unknown function and possibly its derivatives at a single point. Under suitable conditions it has a unique solution.

An equation in which the unknown function and its derivatives appear only to the first power and are not multiplied together. Superposition applies to solutions of the associated homogeneous linear equation; a nonhomogeneous general solution is a particular solution plus a homogeneous one.

The algebraic equation obtained by substituting an exponential trial solution into a linear constant-coefficient differential equation. Its roots determine the solution's form and behavior.

The property that solutions of a system remain bounded or return to an equilibrium after small perturbations. For linear systems it is decided by the signs of eigenvalue real parts.

An integral transform converting a function of time into a function of a complex frequency variable. It turns linear differential equations into algebraic equations solved by partial fractions.