MA1034: Process Modelling using Linear Algebra

Session 02: Generating Sets & Dimension

Prof. Alejandro Ucan-Puc

Tecnológico de Monterrey • Departamento de Matemáticas

Goals of the Session

  • Understand linear combinations of vectors.
  • Formulate concepts of linear dependence and independence.
  • Define the span (generated space) of a set of vectors.
  • Define a basis and the dimension of a vector space.

Motivation: Workstation Production Constraints

An industrial production system has 4 workstations (A, B, C, D) sharing raw materials and labor:

  • Outputs of workstations A and B equal outputs of C and D ($x_A + x_B = x_C + x_D$).
  • Workstation D requires twice the labor of workstation A ($x_D = 2 x_A$).

What is the dimension of feasible output vectors $(x_A, x_B, x_C, x_D) \in \mathbb{R}^4$?

Linear Combinations

Definition: Let $v_1, v_2, \dots, v_k \in V$. A vector $v \in V$ is a linear combination of $\{v_1, \dots, v_k\}$ if there exist scalars $c_1, c_2, \dots, c_k \in \mathbb{R}$ such that:

$$v = c_1 v_1 + c_2 v_2 + \dots + c_k v_k$$

The Span (Generated Space)

Definition: The span of a set $S = \{v_1, \dots, v_k\}$, denoted $\text{Span}(S)$, is the set of all possible linear combinations of vectors in $S$:

$$\text{Span}(S) = \{ c_1 v_1 + \dots + c_k v_k \mid c_i \in \mathbb{R} \}$$

Theorem: $\text{Span}(S)$ is always a subspace of $V$.

Linear Dependence & Independence

Definition: A set of vectors $\{v_1, v_2, \dots, v_k\}$ is linearly independent if the equation:

$$c_1 v_1 + c_2 v_2 + \dots + c_k v_k = 0$$

has only the trivial solution $c_1 = c_2 = \dots = c_k = 0$.

If there exist non-zero scalars $c_i \neq 0$ satisfying the equation, the set is linearly dependent.

Basis of a Vector Space

Definition: A subset $B = \{v_1, v_2, \dots, v_n\} \subset V$ is a basis for $V$ if:

  1. $B$ is linearly independent.
  2. $B$ spans $V$ (i.e., $\text{Span}(B) = V$).

Dimension

Definition: The dimension of a finite-dimensional vector space $V$, denoted $\text{dim}(V)$, is the number of vectors in any basis of $V$.

  • $\text{dim}(\mathbb{R}^n) = n$ (Canonical basis $\{e_1, \dots, e_n\}$).
  • $\text{dim}(P_n(\mathbb{R})) = n + 1$ (Basis $\{1, t, t^2, \dots, t^n\}$).

Summary & Next Step

Bases provide unique coordinate systems for vector spaces, defining the degrees of freedom in engineering models.

Next Session: Linear Transformations, Kernels, and Images.