MATH25a: Second Midterm Definitions

Invertible

Isomorphisms

Homomorphism & automorphism

Definition of matrix

Operator

Product space

Affine subset

Quotient space

Linear functional

Riesz representation theorem

Dual space

Dual map

Transpose

Invariant subspace

Eigenvalue

Eigenvector

Restriction operator

GLn(F)

set of all invertible nxn matrices

Upper-triangular matrix

Trace

Determinant

Characteristic equation

det(A - \lambda I) = 0

Diagonal matrix

Eigenspace

Diagonalizable

Conditions equivalent to diagonalizable

Euclidean inner product

Properties of inner products & what it is

Positivity, definiteness, linearity in the first argument, conjugate symmetry, polarization identity, parallelogram identity

Inner product space

Properties of norms & what it means

Positivity, definiteness, scalar multiplication, triangle inequality

Orthogonal

if inner product equals zero

Pythagorean theorem

if two vectors are orthogonal, then |v+w|² = |v|² + |w|²

Induced norm & how to check if a norm is induced

Cauchy-Schwarz inequality

Orthonormal + orthonormal basis

Fourier decomposition with orthonormal basis

Gram-Schmidt algorithm

Orthogonal complement

Orthogonal projection

Minimizing distance to subspace

Adjoint + properties

Kernel and range of adjoint

Self-adjoint

Normal operator

Complex Spectral Theorem

Real Spectral Theorem

Positive operator

Square root of an operator

Isometry

what does S and S^-1 means in A=SDS^-1 if A is diagonalizable

Properties of determinants

linearity of columns, scalar multiplication

Elementary matrices

Cofactor expansion

Block matrices

Geometric multiplicity

Algebraic multiplicity

Conjugate transpose