5 Vector Space Rn

Vector Space Rn

Introduction

  • We investigate the set Rn of all n-tuples, which are referred to as vectors.

  • Matrix transformations affecting vectors in Rn are given by multiplying by an m × n matrix.

  • Previous sections have explored geometric transformations like rotations and reflections in R2 and R3, specifically in relation to linear transformations and determinants.

Subspaces of Rn

Definition

  • A set U of vectors in Rn is a subspace if it satisfies:

    • S1: The zero vector 0 is in U.

    • S2: If x and y are in U, then x + y is also in U.

    • S3: If x is in U, then ax is in U for every real number a.

Key Points

  • Rn is a subspace of itself.

  • The set U = {0} (zero subspace) is the only subspace containing only the zero vector.

  • Proper subspaces are those which are neither {0} nor Rn.

Properties of Subspaces

  • Closure under addition (S2) and scalar multiplication (S3) are essential properties of subspaces.

  • Example subspaces include any line or plane through the origin in Rn.

Null Space and Image Space of a Matrix

  • The null space of a matrix A, denoted null A, is:

    • null A = {x in Rn | Ax = 0}.

  • The image space of A, denoted im A, is:

    • im A = {Ax | x in Rn}.

  • Both null A and im A are subspaces of Rn or Rm.

Example of Subspaces

  1. Example 5.1.2: Null A is a subspace of Rn because:

    • It contains the zero vector.

    • It is closed under addition and scalar multiplication.

  2. Eigenspaces: Eλ(A) = {x in Rn | Ax = λx} is a subspace related to eigenvalues of matrix A.

    • Eλ(A) correlates to eigenvalues with dimension equal to the number of linearly independent eigenvectors.

Spanning Sets

  • A spanning set for a subspace consists of vectors such that every vector in the subspace can be expressed as a linear combination of those vectors.

  • Example 5.2: The span of vectors can represent any line or plane in Rn based on linearly combining those vectors.

Properties of Spanning Sets

  • The span of a single vector is the line through the origin in the direction of that vector.

  • Theorem 5.1.1 states that the span of a set of vectors is the smallest subspace containing those vectors.

Independence and Dimension

  1. Definition of Linear Independence: A set of vectors is linearly independent if the only solution to t1x1 + t2x2 + ... + tkxk = 0 is t1 = t2 = ... = tk = 0.

  2. Dimension: The dimension of a subspace is the number of vectors in any basis for that subspace, ensuring all vectors are independent.

  3. Concept of Basis: A minimal spanning set can also be thought of as a basis, meaning it spans the space but contains no redundant vectors.

Best Approximation and Least Squares

  • In practical applications, finding solutions that minimize deviations is often crucial, especially for over-determined systems where the number of equations exceeds unknowns.

Summary of Key Concepts

  • The methods of linear algebra provide tools for exploring the relationships between vectors, subspaces, independence, and matrix transformations.

  • Understanding vector spaces and their properties is central to further studying linear algebra, particularly in applications involving transformations, eigenvalues, and approximations.