Mathematics Foundations/6.1 Vectors and Vector Spaces

6.1 Vectors and Vector Spaces 6.1.1 Introduction to Vectors A vector is a mathematical object that has both magnitude and direction. Vectors are fundamental to linear algebra and have numerous applications in physics, engineering, computer science, and other fields. 6.1.1.1 Geometric Representation In two or three dimensions, vectors can be visualized as directed line segments (arrows) in a coordinate system. A vector from point to point is denoted as . 6.1.1.2 Algebraic Representation Vectors can be represented algebraically as ordered tuples of numbers. In , an n-dimensional vector is written as: or where are the components of the vector. 6.1.2 Vector Operations 6.1.2.1 Vector Addition Given two vectors and , their sum is: Geometrically, vector addition follows the parallelogram law. 6.1.2.2 Scalar Multiplication For a scalar and a vector , scalar multiplication gives: Geometrically, scalar multiplication changes the length of a vector (and direction if ). 6.1.2.3 Dot Product The dot product (or inner product) of two vectors and is: Geometrically, , where is the angle between the vectors. Properties of the dot product: Commutativity: Distributivity: Scalar multiplication: 6.1.2.4 Cross Product The cross product is defined only for three-dimensional vectors. For and : Geometrically, , and the direction is perpendicular to both vectors following the right-hand rule. Properties of the cross product: Anticommutativity: Distributivity: Scalar multiplication: 6.1.2.5 Vector Magnitude The magnitude (or length) of a vector is: A unit vector has magnitude 1. Any non-zero vector can be normalized to a unit vector: 6.1.3 Vector Spaces 6.1.3.1 Definition A vector space over a field (typically or ) is a set equipped with two operations: vector addition and scalar multiplication, satisfying the following axioms: For all and all scalars : 1. Closure under addition: 2. Commutativity of addition: 3. Associativity of addition: 4. Additive identity: There exists such that for all 5. Additive inverse: For each , there exists such that 6. Closure under scalar multiplication: 7. Distributivity of scalar multiplication with respect to vector addition: 8. Distributivity of scalar multiplication with respect to field addition: 9. Associativity of scalar multiplication: 10. Scalar multiplication identity: 6.1.3.2 Examples of Vector Spaces : The space of all n-tuples of real numbers : The space of all n-tuples of complex numbers : The space of all polynomials with degree at most n : The space of all matrices : The space of all continuous functions on the interval 6.1.4 Subspaces A subspace of a vector space is a subset of that is itself a vector space under the same operations. 6.1.4.1 Subspace Test A non-empty subset of a vector space is a subspace if and only if: 1. Closure under addition: For all , 2. Closure under scalar multiplication: For all and all scalars , Note: The zero vector must be in every subspace. 6.1.4.2 Examples of Subspaces The set of all vectors in that lie in a plane through the origin The set of all vectors in that lie on a line through the origin The set of all symmetric matrices The set of all polynomials in with zero constant term 6.1.5 Linear Combinations and Span 6.1.5.1 Linear Combination A linear combination of vectors is an expression of the form: where are scalars. 6.1.5.2 Span The span of a set of vectors , denoted , is the set of all possible linear combinations of these vectors: The span of any set of vectors is always a subspace. 6.1.6 Linear Independence 6.1.6.1 Definition A set of vectors is linearly independent if the equation: has only the trivial solution . If there exists a non-trivial solution, the vectors are linearly dependent. 6.1.6.2 Geometric Interpretation In , two vectors are linearly independent if they don't lie on the same line through the origin. In , three vectors are linearly independent if they don't lie on the same plane through the origin. 6.1.7 Basis and Dimension 6.1.7.1 Basis A basis for a vector space is a linearly independent set of vectors that spans . Properties of a basis: Every vector in can be expressed uniquely as a linear combination of the basis vectors. If is a basis, then any vector can be written as: where the scalars are uniquely determined. 6.1.7.2 Standard Basis The standard basis for consists of the vectors: 6.1.7.3 Dimension The dimension of a vector space , denoted , is the number of vectors in any basis for . Properties: All bases for a given vector space have the same number of elements. (the basis is ) 6.1.8 Coordinates and Change of Basis 6.1.8.1 Coordinates Given a basis for a vector space , the coordinates of a vector with respect to are the coefficients such that: The coordinate vector of with respect to is: 6.1.8.2 Change of Basis Let and be two bases for a vector space . The change of basis matrix transforms coordinates with respect to to coordinates with respect to : The columns of are the coordinate vectors of the basis vectors of with respect to : 6.1.9 Exercises 1. Prove that the set of all vectors satisfying for a fixed non-zero vector forms a subspace of . 2. Determine whether the vectors , , and are linearly independent. 3. Find a basis for the subspace of spanned by the vectors , , , and . 4. Let be a basis for . Find the coordinate vector of with respect to . 5. Find the change of basis matrix from the standard basis of to the basis .