➗Linear Algebra and Differential Equations Unit 3 – Vector Spaces

What's a Vector Space?

• Algebraic structure consisting of a set of elements called vectors and two operations: vector addition and scalar multiplication
• Vectors can be added together and multiplied by scalars (real or complex numbers) to produce another vector in the space
• Must satisfy certain axioms (properties) to qualify as a vector space
• Generalizes the notion of Euclidean vectors (e.g., 2D or 3D vectors) to higher dimensions and abstract settings
• Examples include:
  • $\mathbb{R}^n$: n-dimensional real vector space
  • $\mathbb{C}^n$: n-dimensional complex vector space
  • Space of polynomials of degree $\leq n$
  • Space of continuous functions on an interval

Key Properties and Axioms

• Closure under vector addition: If $\vec{u}$ and $\vec{v}$ are vectors in the space, then $\vec{u} + \vec{v}$ is also a vector in the space
• Closure under scalar multiplication: If $\vec{v}$ is a vector and $c$ is a scalar, then $c\vec{v}$ is also a vector in the space
• Associativity of vector addition: $(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$
• Commutativity of vector addition: $\vec{u} + \vec{v} = \vec{v} + \vec{u}$
• Existence of zero vector: There exists a unique vector $\vec{0}$ such that $\vec{v} + \vec{0} = \vec{v}$ for all vectors $\vec{v}$
• Existence of additive inverse: For every vector $\vec{v}$, there exists a unique vector $-\vec{v}$ such that $\vec{v} + (-\vec{v}) = \vec{0}$
• Distributivity of scalar multiplication over vector addition: $c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}$
• Distributivity of scalar multiplication over field addition: $(c + d)\vec{v} = c\vec{v} + d\vec{v}$

Types of Vector Spaces

• Real vector spaces: Vectors with real number components and scalars from the real number field
  • Examples: $\mathbb{R}^2$ (2D plane), $\mathbb{R}^3$ (3D space)
• Complex vector spaces: Vectors with complex number components and scalars from the complex number field
  • Example: $\mathbb{C}^3$ (3D space with complex components)
• Function spaces: Vectors are functions and operations are pointwise
  • Examples: space of continuous functions, space of differentiable functions
• Polynomial spaces: Vectors are polynomials and operations are polynomial addition and scalar multiplication
  • Example: space of polynomials of degree $\leq 3$
• Matrix spaces: Vectors are matrices and operations are matrix addition and scalar multiplication
  • Example: space of $2 \times 2$ real matrices

Subspaces and Span

• Subspace: A subset of a vector space that is itself a vector space under the same operations
  • Must contain the zero vector and be closed under vector addition and scalar multiplication
  • Examples: lines and planes passing through the origin in $\mathbb{R}^3$
• Span: The set of all linear combinations of a given set of vectors
  • Linear combination: A sum of scalar multiples of vectors, $c_1\vec{v}_1 + c_2\vec{v}_2 + \cdots + c_n\vec{v}_n$
  • Spanning set: A set of vectors whose span is the entire vector space
  • Examples:
    • Span of $\{(1, 0), (0, 1)\}$ is $\mathbb{R}^2$
    • Span of $\{1, x, x^2\}$ is the space of polynomials of degree $\leq 2$

Linear Independence and Dependence

• Linearly independent: A set of vectors is linearly independent if no vector can be written as a linear combination of the others
  • Equivalently, the only solution to $c_1\vec{v}_1 + c_2\vec{v}_2 + \cdots + c_n\vec{v}_n = \vec{0}$ is $c_1 = c_2 = \cdots = c_n = 0$
  • Example: $\{(1, 0), (0, 1)\}$ is linearly independent in $\mathbb{R}^2$
• Linearly dependent: A set of vectors is linearly dependent if at least one vector can be written as a linear combination of the others
  • Equivalently, there exists a non-trivial solution to $c_1\vec{v}_1 + c_2\vec{v}_2 + \cdots + c_n\vec{v}_n = \vec{0}$
  • Example: $\{(1, 0), (0, 1), (1, 1)\}$ is linearly dependent in $\mathbb{R}^2$ since $(1, 1) = (1, 0) + (0, 1)$
• Importance: Linearly independent sets are crucial for defining bases and dimensions of vector spaces

Basis and Dimension

• Basis: A linearly independent spanning set for a vector space
  • Every vector in the space can be uniquely expressed as a linear combination of basis vectors
  • Examples:
    • Standard basis for $\mathbb{R}^2$: $\{(1, 0), (0, 1)\}$
    • Standard basis for the space of polynomials of degree $\leq 2$: $\{1, x, x^2\}$
• Dimension: The number of vectors in a basis for a vector space
  • All bases for a vector space have the same number of vectors
  • Examples:
    • Dimension of $\mathbb{R}^n$ is $n$
    • Dimension of the space of polynomials of degree $\leq n$ is $n+1$
• Importance: Bases and dimensions provide a way to represent and analyze vector spaces in a standardized manner

Coordinate Systems

• Coordinate system: A way to represent vectors in a vector space using a basis
  • Each vector is identified by a unique tuple of scalars (coordinates) corresponding to the basis vectors
  • Example: In $\mathbb{R}^2$ with standard basis $\{(1, 0), (0, 1)\}$, the vector $(3, 4)$ has coordinates $3$ and $4$
• Change of basis: Expressing vectors in terms of a different basis
  • Useful for simplifying computations or highlighting certain properties of vectors
  • Achieved through matrix multiplication: $[\vec{v}]_B = P[\vec{v}]_A$, where $P$ is the change of basis matrix
• Importance: Coordinate systems allow for the numerical representation and manipulation of vectors in a given basis

Applications in Linear Algebra

• Systems of linear equations: Vector spaces provide a framework for solving and analyzing systems of linear equations
  • Solutions form a subspace (solution space) of the vector space
  • Basis vectors of the solution space give parametric form of solutions
• Linear transformations: Functions between vector spaces that preserve vector addition and scalar multiplication
  • Can be represented by matrices in given bases
  • Eigenvalues and eigenvectors of the matrix provide insights into the transformation's behavior
• Least squares approximation: Finding the best-fitting linear model for a set of data points
  • Minimizes the sum of squared distances between data points and the model
  • Solution involves orthogonal projection onto a subspace spanned by the model's basis functions
• Principal component analysis (PCA): Technique for dimensionality reduction and data compression
  • Identifies the directions (principal components) of maximum variance in the data
  • Projects data onto a lower-dimensional subspace spanned by the principal components