Review of Linear Algebra and Differential Equations

Flashcards containing the vocabulary terms and definitions from the lecture notes.

Eigenvalue

A scalar λ such that Av=λv for some nonzero vector v.

Eigenvector

A nonzero vector v satisfying Av=λv for some scalar λ.

Characteristic equation (of a matrix)

The equation det(A−λI)=0, used to find eigenvalues.

Eigenspace

The span of all eigenvectors corresponding to a given eigenvalue, along with the zero vector.

Similar

Matrices A and B are similar if B = P^{-1}A for some invertible matrix P; they represent the same linear transformation under different bases.

Diagonalizable

A matrix A is diagonalizable if it is similar to a diagonal matrix; this occurs if it has enough linearly independent eigenvectors.

Diagonalization

The process of finding P and D such that A=PDP^−1, where D is diagonal.

Solution of first-order system

A vector-valued function x(t) that satisfies a system x′=Ax+f(t).

Solution curve/trajectory

The path traced by x⃗(t) in space as t varies, representing system behavior.

Homogeneous vs. nonhomogeneous system

A system is homogeneous if x′=Ax; it’s nonhomogeneous if there’s an added f(t), i.e., x′=Ax+f(t).

Matrix(-valued) function

A function that returns a matrix for each value in its domain.

Continuous matrix function

A matrix-valued function whose component functions are all continuous.

Differentiable matrix function

A matrix-valued function whose component functions are all differentiable.

Derivative of a matrix function

A matrix where each entry is the derivative of the corresponding entry in the original matrix function.

Associated homogeneous equation

The homogeneous version of a nonhomogeneous system; e.g., for x′=Ax+f(t), the associated homogeneous equation is x′=Ax.

Linearly independent vs. dependent vector-valued functions

A set of vector functions {x1(t),…,xn(t)} is linearly independent if no nontrivial linear combination equals the zero vector function.

Wronskian

A determinant used to test linear independence of vector-valued functions; for nnn functions, it’s the determinant of a matrix with each function and its derivatives as columns.

Complementary function

The general solution to the homogeneous system; forms part of the full solution to a nonhomogeneous system.

Eigenvalue method

A technique for solving homogeneous linear systems x′=Ax using the eigenvalues and eigenvectors of A.

Multiplicity

The number of times an eigenvalue appears in the characteristic equation (algebraic multiplicity).

Complete

A set of eigenvectors is complete if it spans the space; completeness implies the matrix is diagonalizable.

Defective

A matrix is defective if it does not have enough linearly independent eigenvectors to be diagonalizable.

Defect

The difference between the algebraic and geometric multiplicities of an eigenvalue.

Fundamental matrix

A matrix whose columns are linearly independent solutions to x′=Ax; used to express general solutions.

Method of undetermined coefficients for systems

A technique for solving x′=Ax+f(t) by guessing a particular solution form based on f(t).

Variation of parameters for systems

A general method for finding a particular solution using a fundamental matrix and integration.

Inverse matrix

A matrix A^{-1} such that A*A^{-1} = A^{-1} * A =I, where I is the identity matrix.

Invertible / Nonsingular

A square matrix is invertible (or nonsingular) if its inverse exists; equivalently, its determinant is nonzero.

Elementary matrix

A matrix obtained by performing a single row operation on the identity matrix.

Determinant

A scalar value computed from a square matrix that helps determine invertibility and other properties.

Minor

The determinant of a smaller matrix formed by deleting one row and one column from a square matrix.

Cofactor

A signed minor; used in computing the determinant and inverse of a matrix.

Upper vs. lower triangular

A matrix is upper triangular if all entries below the main diagonal are zero; lower triangular if all above are zero.

Transpose

A matrix obtained by flipping rows and columns; the transpose of A is denoted A^T.

n-dimensional space Rn

The set of all ordered n-tuples of real numbers; standard setting for vector analysis.

Points

Specific locations in Rn, represented by coordinates.

Coordinates

The components of a point or vector, indicating its position relative to the origin.

Vector

An object with both direction and magnitude in Rn, represented as a column of components.

Components of a vector

The individual entries in a vector, typically denoting direction along each axis.

Vector sum

The addition of two vectors by adding corresponding components.

Scalar multiple

A vector scaled by a real number, stretching or shrinking its length.

Length

The magnitude (norm) of a vector V

Vector space

A set of vectors closed under vector addition and scalar multiplication, satisfying 10 axioms.

Zero vector

The unique vector with all components zero; it acts as the additive identity in a vector space.

Basic unit vectors

Vectors in Rn with a 1 in one position and 0 elsewhere, e.g., e1=[1,0,…,0]

Basis

A set of linearly independent vectors that spans a vector space.

Subspace

A subset of a vector space that is itself a vector space under the same operations.

Solution space

The set of all solutions to a linear system or differential equation; forms a subspace if homogeneous.

Nullspace

The set of all solutions Ax=0; also called the kernel.

Zero subspace

The subspace containing only the zero vector.

Proper subspace

A subspace that is strictly contained within a vector space (not equal to the whole space).

Linear combination

A sum of scalar multiples of vectors.

Span

The set of all linear combinations of a given set of vectors.

Spanning set

A set of vectors whose span equals the whole space (or subspace).

Initial condition

A given value of the function (or derivatives) at a specific point.

General vs. particular solution

The general solution contains arbitrary constants; the particular solution satisfies specific initial conditions.

Position function

A function x(t) representing an object’s position over time.

Velocity function

The derivative v(t)=x′(t, representing rate of change of position.

Acceleration function

The derivative a(t)=v′(t)=x′′(t), representing rate of change of velocity.

Solution curve

The graph of a solution function y(t); also called a trajectory.

Slope/direction field

A visual representation of the slope y′y'y′ at points in the plane to show solution behavior.

Separable equation

A first-order ODE that can be solved by separating variables.

Implicit solution

A solution expressed in an equation involving both xxx and yyy, not explicitly solved for y.

General vs. singular solution

A singular solution is not part of the general family but still solves the differential equation.

Dimension

The number of vectors in a basis for a vector space or subspace.

Column space

The subspace of Rm spanned by the columns of a matrix A; also called the range.

Rank

The dimension of the column space (or row space) of a matrix.

Nullity

The dimension of the nullspace of a matrix.

Linear differential equation

A differential equation where the unknown function and its derivatives appear linearly.

Order of a differential equation

The highest derivative of the unknown function that appears in the equation.

Homogeneous linear equation

A linear differential equation where the nonhomogeneous term is zero, e.g., y′′+3y′+2y=0.

Operator

A rule that acts on functions, such as the derivative D=dtd.

Polynomial differential operator

An expression like D^2+3*D+2, acting on a function y(t).

Euler’s Formula

eix=cosx+isinxe^{ix} = \cos x + i \sin xeix=cosx+isinx, used to relate complex exponentials to sines and cosines.

Complex-valued function

A function that outputs complex numbers, often arising in differential equation solutions.

Differential equation

An equation involving a function and its derivatives.

Mathematical model

A mathematical description (often using equations) of a real-world process.

Order (of a differential equation)

The highest derivative present in the equation.

nth-order differential equation

A differential equation involving the nth derivative of the unknown function.

Solution of a differential equation on an interval I

A function that satisfies the differential equation for all t in the interval I.

Ordinary vs. partial differential equation

ODEs involve derivatives with respect to one variable; PDEs involve partial derivatives with respect to multiple variables.

Initial value problem (IVP)

A differential equation plus an initial condition, such as y(t0)=y0.

Decay/growth constant

The constant kkk in equations like y′=ky, indicating rate of exponential growth/decay.

Exponential/natural growth equation

A model of the form y(t)=y0ekt

Half-life

The time it takes for a quantity to decay to half its original value in an exponential decay process.

Integrating factor

A function μ(t) used to solve linear first-order equations by rewriting the equation as a product rule.

Linear first-order equation

An ODE of the form y′+p(t)y=q(t).

General population equation

A differential equation modeling population, such as P′=kPP' = kPP′=kP or logistic growth.

Logistic equation

A population model: P′=kP(1−PL), where L is the carrying capacity.

Limiting population / carrying capacity

The maximum sustainable population in logistic growth.

Threshold population

A value separating different growth behaviors; often associated with unstable/stable points.

Autonomous first-order differential equation

An ODE of the form y′=f(y), where the rate of change depends only on y.

Critical points

Values where y′=0 in an autonomous equation; potential equilibrium points.

Equilibrium solution

A constant solution y(t)=c where y′=0; the system is in steady state.

Phase diagram

A diagram showing the direction of solutions along the yyy-axis for autonomous equations.

Stable vs. unstable critical points

Stable if solutions approach the point; unstable if they move away.

Bifurcation point

A parameter value where the qualitative nature of equilibrium solutions changes.

Bifurcation diagram

A graph showing how equilibrium points vary with a parameter, highlighting bifurcations.

Euler’s method

A numerical method for approximating solutions of ODEs using tangent-line steps.

Error

The difference between the approximate numerical solution and the exact solution.