MA1034: Process Modelling using Linear Algebra

Session 01: Vector Spaces

Prof. Alejandro Ucan-Puc

Tecnológico de Monterrey • Departamento de Matemáticas

Goals of the Session

  • Define the formal algebraic structure of a vector space.
  • Analyze key examples and counterexamples of vector spaces.
  • Introduce the concept of vector subspaces and their criteria.

Motivation: Production Systems

Three factory workstations share materials to produce products. Each workstation's output depends on materials received, represented by the matrix:

$$T = \begin{pmatrix} 2 & -1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & -1 \end{pmatrix}$$

Columns represent material flows; rows represent workstations. Negative entries represent material consumption.

Key Questions

  1. How to determine when the system does not produce anything?
  2. How can we describe the set of all possible outputs?
  3. Is there a material distribution arrangement that guarantees optimal production?

All these questions are answered by the theory of vector spaces.

What is a Vector Space?

Definition: A set of objects $V$ equipped with two operations (vector addition $+$ and scalar multiplication $\cdot$) is called a vector space over $\mathbb{R}$ if it satisfies the 10 fundamental axioms.

Elements $u, v \in V$ are called vectors, and $\lambda, \mu \in \mathbb{R}$ are called scalars.

Axioms of a Vector Space

  1. Closure under addition: $u + v \in V$ for all $u,v \in V$.
  2. Closure under scalar mult.: $\lambda u \in V$ for all $u \in V, \lambda \in \mathbb{R}$.
  3. Associativity of addition: $(u + v) + w = u + (v + w)$.
  4. Commutativity of addition: $u + v = v + u$.
  5. Additive Identity: $\exists 0 \in V$ such that $u + 0 = u$.
  6. Additive Inverse: $\forall u \in V, \exists -u \in V$ such that $u + (-u) = 0$.
  7. Distributivity 1: $\lambda (u + v) = \lambda u + \lambda v$.
  8. Distributivity 2: $(\lambda + \mu) u = \lambda u + \mu u$.
  9. Associativity of scalar mult.: $(\lambda \mu) u = \lambda (\mu u)$.
  10. Scalar Identity: $1 \cdot u = u$.

Standard Examples

  • Euclidean Space $\mathbb{R}^n$: Tuples of $n$ real numbers with component-wise addition and scalar multiplication.
  • Polynomial Space $P_n(\mathbb{R})$: Polynomials of degree $\le n$ with real coefficients: $$x = a_0 + a_1 t + a_2 t^2 + \dots + a_n t^n$$
  • Function Space $C([a,b])$: Continuous real-valued functions defined on $[a,b]$.

Counterexample

Consider $V = \mathbb{R}^2$ with standard addition, but non-standard scalar multiplication:

$$\lambda (a_1, a_2) = (0, 0)$$

Fails Axiom 10: $1 \cdot (1, 1) = (0, 0) \neq (1, 1)$. Thus, $V$ is NOT a vector space.

Vector Subspaces

Definition: A subset $W \subseteq V$ of a vector space $V$ is a subspace if $W$ is itself a vector space under the operations inherited from $V$.

Subspace Test (3 Criteria):

  1. The zero vector $0 \in W$.
  2. If $u, v \in W$, then $u + v \in W$ (Closed under addition).
  3. If $u \in W$ and $c \in \mathbb{R}$, then $c u \in W$ (Closed under scalar mult.).

Example 1: Production System

An industrial engineer studies a production system with 4 workstations $A, B, C, D$ whose outputs are represented by a vector

$$x = (x_A, x_B, x_C, x_D) \in \mathbb{R}^4.$$

The system is constrained by:

  • Material balance: $x_A + x_B = x_C + x_D$
  • Labor constraint: $x_D = 2x_A$

Substituting the second relation into the first gives

$$x_A + x_B = x_C + 2x_A \quad \Rightarrow \quad x_B = x_A + x_C.$$

Hence every feasible output combination is of the form

$$ (x_A, x_B, x_C, x_D) = (x_A, x_A + x_C, x_C, 2x_A) = x_A(1,1,0,2) + x_C(0,1,1,0). $$

So the set of all feasible output combinations is a 2-dimensional subspace of $\mathbb{R}^4$.

Example 2: Robotic Arm

A mechatronics engineer designs a robotic arm with three joint angles $(\theta_A, \theta_B, \theta_C) \in \mathbb{R}^3$.

The end-effector position is constrained by the feasibility condition

$$\theta_A + \theta_B + \theta_C = 2\pi.$$

This means the admissible joint configurations form the plane

$$\{(\theta_A, \theta_B, \theta_C) \in \mathbb{R}^3 : \theta_A + \theta_B + \theta_C = 2\pi\}.$$

In geometric terms, this is a 2-dimensional affine plane in $\mathbb{R}^3$; it is described by a linear equation and therefore has dimension 2.

Summary & Next Step

Vector spaces provide the rigorous framework to study linear equations, production systems, and process optimization.

Next Session: Generating Sets, Linear Independence, and Dimension.