MATHEMATICS S5 UNIT 6: Vector Space of Real Numbers.

A “Vector Space of Real Numbers” course is typically a foundational course in Linear Algebra. It moves beyond the concrete geometric vectors you might have encountered in physics or introductory calculus (like arrows in 2D or 3D space) to a more abstract and generalized concept of a “vector.” The “real numbers” part signifies that the scalars (the numbers you multiply vectors by) are real numbers, distinguishing it from complex vector spaces where scalars can be complex numbers.

Here’s a breakdown of what such a course typically covers:

Core Concepts and Content:

1. Definition of a Vector Space:

   • This is the cornerstone of the course. Students learn the formal definition of a vector space as a set V equipped with two operations: vector addition and scalar multiplication, which must satisfy a specific list of ten axioms. These axioms generalize the familiar properties of vector arithmetic (e.g., commutativity and associativity of addition, existence of a zero vector and additive inverses, distributive properties, etc.).
   • Emphasis is placed on understanding why these axioms are important and how they define the structure of a vector space.

2. Examples of Vector Spaces:

   • Beyond the intuitive Rⁿ (n-tuples of real numbers like vectors in 2D or 3D space), the course explores various non-traditional examples that satisfy the vector space axioms. These often include:
     • Spaces of polynomials (e.g.,  P_n​, the set of all polynomials of degree less than or equal to $n$).
     • Spaces of matrices (e.g., M_m×n​, the set of all $m\times n$ matrices).
     • Spaces of continuous functions.
     • The trivial vector space (containing only the zero vector).

3. Subspaces:

   • Definition of a subspace: A subset of a vector space that is itself a vector space under the same operations.
   • How to check if a subset is a subspace (typically by verifying closure under addition and scalar multiplication, and containing the zero vector).
   • Examples of subspaces (e.g., lines or planes through the origin in R³, solution sets to homogeneous linear systems).

4. Linear Combinations, Span, and Spanning Sets:

   • Linear Combination: Expressing one vector as a sum of scalar multiples of other vectors.
   • Span: The set of all possible linear combinations of a given set of vectors. This forms a subspace.
   • Spanning Set: A set of vectors whose span is the entire vector space (or a subspace).

5. Linear Independence and Dependence:

   • Linear Independence: A set of vectors where no vector can be written as a linear combination of the others (i.e., the only way to get the zero vector as a linear combination is if all scalar coefficients are zero).
   • Linear Dependence: A set of vectors where at least one vector can be written as a linear combination of the others.
   • Techniques for determining linear independence (e.g., setting up a homogeneous system of linear equations and checking for non-trivial solutions).

6. Basis and Dimension:

   • Basis: A linearly independent set of vectors that also spans the entire vector space. A basis provides a “minimal” set of building blocks for the space.
   • Dimension: The number of vectors in any basis for a given vector space. This concept formalizes the intuitive idea of “n-dimensional space.”
   • Standard bases (e.g., the standard basis for R_n).
   • Finding a basis for a given vector space or subspace.

7. Coordinate Systems:

   • Using a basis to represent any vector in the space as a unique set of coordinates relative to that basis.
   • Change of basis (transforming coordinates from one basis to another).