Abstract Linear Algebra I Unit 1 Review: Vector Spaces and Subspaces

Key Concepts and Definitions

• Vector spaces consist of a set of vectors and two operations (vector addition and scalar multiplication) that satisfy certain axioms
• Vectors are elements of a vector space represented as ordered tuples or arrays of numbers (components)
• Scalars are elements of a field (real numbers or complex numbers) used to scale vectors through multiplication
• Zero vector is a unique vector in every vector space where adding it to any other vector results in the original vector
• Linear combination expresses a vector as the sum of scalar multiples of other vectors
• Span is the set of all possible linear combinations of a given set of vectors
• Linear independence means a set of vectors cannot be expressed as linear combinations of each other
  • Linearly dependent vectors have at least one vector that can be written as a linear combination of the others
• Basis is a linearly independent set of vectors that spans the entire vector space
• Dimension is the number of vectors in a basis for a vector space

Vector Space Axioms

• Closure under addition states that the sum of any two vectors in the space results in another vector within the space
• Closure under scalar multiplication ensures that multiplying a vector by a scalar yields a vector still in the space
• Associativity of addition: $(u + v) + w = u + (v + w)$ for any vectors $u$, $v$, and $w$ in the space
• Commutativity of addition: $u + v = v + u$ for any vectors $u$ and $v$ in the space
• Additive identity: There exists a unique zero vector $0$ such that $v + 0 = v$ for any vector $v$ in the space
• Additive inverses: For every vector $v$ in the space, there exists a unique vector $-v$ such that $v + (-v) = 0$
• Distributivity of scalar multiplication over vector addition: $a(u + v) = au + av$ for any scalar $a$ and vectors $u$ and $v$
• Distributivity of scalar multiplication over field addition: $(a + b)v = av + bv$ for any scalars $a$ and $b$ and vector $v$

Examples of Vector Spaces

• The set of all $n$-tuples of real numbers ($\mathbb{R}^n$) with component-wise addition and scalar multiplication
• The set of all polynomials with real coefficients ($\mathbb{P}$) with polynomial addition and scalar multiplication
• The set of all $m \times n$ matrices with real entries ($\mathbb{M}_{m,n}(\mathbb{R})$) with matrix addition and scalar multiplication
• The set of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$ ($C(\mathbb{R})$) with point-wise addition and scalar multiplication
• The set of all solutions to a homogeneous linear differential equation with constant coefficients
  • Example: The set of functions of the form $ae^{2t} + be^{-3t}$ where $a$ and $b$ are real numbers

Subspaces and Their Properties

• A subspace is a subset of a vector space that is itself a vector space under the same operations
• The zero vector of the parent space must be included in the subspace
• Closure under addition: The sum of any two vectors in the subspace must also be in the subspace
• Closure under scalar multiplication: Multiplying any vector in the subspace by a scalar must result in a vector within the subspace
• Examples of subspaces include any line or plane passing through the origin in $\mathbb{R}^3$
• The set of all polynomials of degree at most $k$ is a subspace of $\mathbb{P}$
• The set of all symmetric $n \times n$ matrices is a subspace of $\mathbb{M}_{n,n}(\mathbb{R})$
• The intersection of any two subspaces of a vector space is also a subspace

Span and Linear Combinations

• The span of a set of vectors $\{v_1, v_2, \ldots, v_k\}$ is the set of all linear combinations of these vectors
  • Denoted as $\text{Span}(v_1, v_2, \ldots, v_k) = \{a_1v_1 + a_2v_2 + \ldots + a_kv_k : a_1, a_2, \ldots, a_k \in \mathbb{R}\}$
• The span of a set of vectors is always a subspace of the vector space containing the vectors
• A vector space is spanned by a set of vectors if every vector in the space can be expressed as a linear combination of the vectors in the set
• The span of the standard basis vectors $\{e_1, e_2, \ldots, e_n\}$ in $\mathbb{R}^n$ is the entire space $\mathbb{R}^n$
• The span of the monomials $\{1, x, x^2, \ldots, x^n\}$ is the set of all polynomials of degree at most $n$

Linear Independence and Dependence

• A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the other vectors
  • Equivalently, the only solution to $a_1v_1 + a_2v_2 + \ldots + a_kv_k = 0$ is $a_1 = a_2 = \ldots = a_k = 0$
• A set of vectors is linearly dependent if at least one vector can be expressed as a linear combination of the others
• The zero vector is always linearly dependent on any set of vectors
• Any set containing the zero vector is linearly dependent
• The standard basis vectors $\{e_1, e_2, \ldots, e_n\}$ in $\mathbb{R}^n$ are linearly independent
• The monomials $\{1, x, x^2, \ldots, x^n\}$ are linearly independent in the vector space of polynomials

Basis and Dimension

• A basis for a vector space is a linearly independent set of vectors that spans the entire space
• The number of vectors in a basis is the dimension of the vector space
• All bases for a vector space have the same number of vectors (i.e., the dimension is unique)
• The standard basis for $\mathbb{R}^n$ is $\{e_1, e_2, \ldots, e_n\}$, where $e_i$ has a 1 in the $i$-th position and 0s elsewhere
• The standard basis for the space of polynomials of degree at most $n$ is $\{1, x, x^2, \ldots, x^n\}$
• The dimension of the zero vector space (containing only the zero vector) is 0
• The dimension of $\mathbb{R}^n$ is $n$
• The dimension of the space of polynomials of degree at most $n$ is $n+1$

Applications and Problem-Solving Techniques

• Determine if a given set is a vector space by verifying the vector space axioms
• Check if a subset of a vector space is a subspace by testing closure under addition and scalar multiplication
• Express a vector as a linear combination of other vectors by solving a system of linear equations
• Determine if a set of vectors spans a vector space by checking if every vector in the space can be written as a linear combination of the set
• Verify linear independence of a set of vectors by solving the equation $a_1v_1 + a_2v_2 + \ldots + a_kv_k = 0$ and checking if the only solution is the trivial solution
• Find a basis for a vector space by starting with a spanning set and removing linearly dependent vectors until a linearly independent spanning set remains
• Compute the dimension of a vector space by finding the number of vectors in a basis
• Use the concept of linear independence to solve problems related to systems of linear equations and matrix equations