Vector Subspaces, Linear Applications, and Quadratic Forms

Posted on Mar 4, 2025 in Geology

A.1.7. Vector Subspace

Let S be a subset of a vector space E. S is a vector subspace if and only if:

  • It contains the zero vector (0).
  • It simultaneously meets these conditions:
    • For any two vectors v1 and v2 belonging to S, v1 + v2 also belongs to S.
    • For any vector v1 and any real number k, kv1 belongs to S.

The following are not vector subspaces:

  • Numbers with exponents.
  • Vectors defined by u = (x, y) where xy + x = 0 (product of two coordinates).
  • Logarithms, etc. (certain mathematical constructs).

The following are vector subspaces:

  • Ax + By + Cz = 0
  • C…x + …y = 0

A.4.2. Kernel and Image of a Linear Application

Let f be a linear application: E → F. The kernel of a linear application f, denoted as ker(f), is the set formed by all vectors whose image through f is the zero vector. That is, all vectors whose image is 0. The set ker(f) is a vector subspace of E.

The image of a linear application f, denoted as im(f), is the set formed by all vectors in F that are the image of some element in E. im(f) is a vector subspace of F. The dimension of the image coincides with the rank of the matrix associated with f:

dim(im(f)) = rg(m(f))

Furthermore:

  • f is injective ⇔ ker(f) = {0}
  • f is surjective ⇔ rg(m(f)) = dim(F)
  • Important: dim(ker(f)) + dim(im(f)) = dim(E)
  • If E = F and f is injective ⇒ f is bijective
  • If E = F and f is surjective ⇒ f is bijective

A.5.1. Eigenvalues and Eigenvectors

A real number λ is an eigenvalue of a matrix if there exists a non-zero vector x such that:

Ax = λx

If λ is an eigenvalue, then the non-zero vectors that satisfy the above relation are called eigenvectors associated with λ.

Each eigenvector is associated with a unique eigenvalue. However, each eigenvalue can have one or more associated eigenvectors.

A.6.1. Quadratic Form

A quadratic form q on a vector space Rn is an application:

q: Rn → R

x → q(x) = a (where a is a real number)

It satisfies the property that q(λx) = λ2q(x), for any vector x in Rn and any real number λ.

It also holds that q(0) = 0.

Classification of quadratic forms:

  • Positive Definite (PD): if q(x) > 0
  • Positive Semidefinite (PSD): if q(x) ≥ 0
  • Negative Definite (ND): if q(x) < 0
  • Negative Semidefinite (NSD): if q(x) ≤ 0
  • Indefinite: If it has no defined sign, that is, if q(x) can be both positive and negative (if it is not any of the four previous cases).

C.1.1. Level Curves

Level curves are the set of all points where the function has the same value. That is, given a function f(x, y), set it equal to a real value, solve for y as a function of x, and graph it.

C.1.2. Continuity

  • For a function f(x):

    f(x) is continuous at a point a if the limit of f(x) as x approaches a from the left and from the right coincides with the image of a.

  • For a function f(x, y):

    f(x, y) is continuous at a point (x0, y0) if the limit of f(x,y) as (x,y) approaches (x0, y0) coincides with the image of (x0, y0).

Properties of continuity:

  • All elementary functions (polynomials, sine, cosine, exponential, logarithmic, etc.) are continuous throughout their domain.
  • Elementary functions related by addition, subtraction, multiplication, or division result in another continuous function.