➗Abstract Linear Algebra II Unit 1 – Vector Spaces and Subspaces

Key Concepts and Definitions

• Vector space consists of a set $V$ and two operations (addition and scalar multiplication) satisfying specific axioms
• Subspace is a subset of a vector space that is closed under addition and scalar multiplication
• Linear combination expresses a vector as the sum of scalar multiples of other vectors
• Span of a set of vectors is the set of all linear combinations of those vectors
• Linearly independent set has no vector that can be expressed as a linear combination of the others
• Linearly dependent set contains at least one vector that is a linear combination of the others
• Basis is a linearly independent set that spans the entire vector space
• Dimension of a vector space is the number of vectors in any basis for that space

Vector Space Fundamentals

• Vector space axioms include closure, associativity, commutativity, identity, inverse, and distributivity properties
• Examples of vector spaces include $\mathbb{R}^n$, $\mathbb{C}^n$, and the set of all polynomials of degree $\leq n$
• Zero vector serves as the additive identity in a vector space
• Scalar multiplication distributes over vector addition $a(\vec{u} + \vec{v}) = a\vec{u} + a\vec{v}$
• Vector addition distributes over scalar addition $(a + b)\vec{v} = a\vec{v} + b\vec{v}$
• Scalar multiplication is compatible with field multiplication $a(b\vec{v}) = (ab)\vec{v}$
• Multiplicative identity property states $1\vec{v} = \vec{v}$ for all $\vec{v}$ in the vector space

Subspace Theory

• Subspace inherits the vector space structure from its parent space
• Trivial subspaces include the zero vector space $\{0\}$ and the entire vector space $V$
• Sum of two subspaces $U + W = \{\vec{u} + \vec{w} : \vec{u} \in U, \vec{w} \in W\}$ is also a subspace
  • Example: even functions and odd functions form subspaces of the vector space of all functions
• Intersection of two subspaces $U \cap W = \{\vec{v} : \vec{v} \in U \text{ and } \vec{v} \in W\}$ is also a subspace
• Direct sum of two subspaces $U \oplus W$ requires $U \cap W = \{0\}$
• Subspace test verifies closure under addition and scalar multiplication
  • Example: set of all polynomials with real coefficients of degree $\leq 2$ is a subspace of $\mathbb{R}[x]$

Linear Combinations and Span

• Linear combination of vectors $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n$ is $a_1\vec{v}_1 + a_2\vec{v}_2 + \cdots + a_n\vec{v}_n$, where $a_1, a_2, \ldots, a_n$ are scalars
• Span of a set of vectors $S = \{\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n\}$ is denoted as $\text{span}(S)$ or $\text{span}(\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n)$
  • Example: $\text{span}((1, 0), (0, 1)) = \mathbb{R}^2$
• Spanning set for a vector space $V$ is a set of vectors whose span is equal to $V$
• Column space of a matrix $A$ is the span of its column vectors
• Row space of a matrix $A$ is the span of its row vectors

Linear Independence and Dependence

• Linearly independent set has no vector that can be expressed as a linear combination of the others
  • Example: standard basis vectors $\vec{e}_1, \vec{e}_2, \ldots, \vec{e}_n$ are linearly independent
• Linearly dependent set contains at least one vector that is a linear combination of the others
• Unique representation property states that every vector in a linearly independent set has a unique representation as a linear combination of the set's vectors
• Linear dependence lemma states that a set $\{\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n\}$ is linearly dependent if and only if some $\vec{v}_i$ is a linear combination of the preceding vectors $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_{i-1}$
• Linearly dependent set cannot be a basis for a vector space

Basis and Dimension

• Basis is a linearly independent set that spans the entire vector space
  • Example: standard basis for $\mathbb{R}^n$ is $\{\vec{e}_1, \vec{e}_2, \ldots, \vec{e}_n\}$, where $\vec{e}_i$ has a 1 in the $i$-th position and 0s elsewhere
• Dimension of a vector space is the number of vectors in any basis for that space
• Finite-dimensional vector space has a finite basis, while an infinite-dimensional vector space has an infinite basis
• Coordinate vector of $\vec{v}$ with respect to a basis $B$ is the unique linear combination of basis vectors that equals $\vec{v}$
• Change of basis matrix transforms coordinates from one basis to another
• Rank-nullity theorem states that for a linear map $T: V \to W$, $\dim(V) = \dim(\text{null}(T)) + \dim(\text{range}(T))$

Applications and Examples

• Function spaces, such as the space of continuous functions $C[a, b]$ or the space of square-integrable functions $L^2[a, b]$, are infinite-dimensional vector spaces
• Polynomial vector spaces $\mathbb{R}[x]$ or $\mathbb{C}[x]$ have bases consisting of monomials $\{1, x, x^2, \ldots\}$
• Solution space of a homogeneous linear system $A\vec{x} = \vec{0}$ is a subspace of $\mathbb{R}^n$ (or $\mathbb{C}^n$)
  • Example: solution space of $x_1 + 2x_2 - x_3 = 0$ is a plane through the origin in $\mathbb{R}^3$
• Eigen spaces corresponding to eigenvalues of a linear operator are subspaces of the domain
• Kernel (null space) and image (range) of a linear transformation are subspaces of the domain and codomain, respectively

Common Pitfalls and Tips

• Subspace test requires closure under both addition and scalar multiplication, not just one or the other
• Linear independence is not the same as orthogonality; vectors can be linearly independent without being orthogonal
• Basis vectors are not unique; a vector space can have multiple bases
• Dimension is a property of the vector space, not the choice of basis
• Coordinate vectors depend on the choice of basis; changing the basis changes the coordinates
• Linearly dependent set cannot span the entire vector space, as it will miss some vectors
• Linearly independent spanning set is a basis; linearly independent non-spanning set is not a basis
• When working with abstract vector spaces, focus on the properties and axioms rather than the specific nature of the elements