1.2 Subspaces and Span

Subspaces and span are key concepts in understanding vector spaces. They help us break down complex spaces into simpler parts and build them up from basic components. These ideas are crucial for grasping how vectors interact and combine within a larger structure.

By exploring subspaces and span, we can see how smaller vector spaces fit inside larger ones. This knowledge lets us solve problems more easily by working with simpler pieces instead of tackling the whole space at once.

Subspaces of Vector Spaces

Definition and Properties

• A subspace is a subset of a vector space that satisfies three conditions:
  • It contains the zero vector
  • It is closed under vector addition (if $u$ and $v$ are in the subspace, then $u + v$ is also in the subspace)
  • It is closed under scalar multiplication (if $v$ is in the subspace and $c$ is a scalar, then $cv$ is also in the subspace)
• If $W$ is a subspace of a vector space $V$, then $W$ is itself a vector space under the same operations as $V$ (addition and scalar multiplication)
• The empty set and the vector space $V$ itself are always subspaces of $V$, known as the trivial subspaces

Intersection and Examples

• The intersection of any collection of subspaces of a vector space $V$ is also a subspace of $V$
  • For example, if $W_1$ and $W_2$ are subspaces of $V$, then $W_1 \cap W_2$ is also a subspace of $V$
• The set of all solutions to a homogeneous linear system of equations forms a subspace of the vector space $\mathbb{R}^n$ or $\mathbb{C}^n$, where $n$ is the number of variables in the system
• The set of all polynomials of degree at most $n$ forms a subspace of the vector space of all polynomials

Identifying Subspaces

Proving a Subset is a Subspace

• To prove that a subset $W$ of a vector space $V$ is a subspace, one must show that $W$ satisfies the three conditions of a subspace:
  1. It contains the zero vector
  2. It is closed under vector addition
  3. It is closed under scalar multiplication
• If a subset $W$ of a vector space $V$ fails to satisfy any one of the three conditions, then $W$ is not a subspace of $V$

Examples of Subspaces and Non-Subspaces

• The set of all matrices with real entries is a subspace of the vector space of all matrices with complex entries
• The set of all continuous functions on the interval $[0, 1]$ is a subspace of the vector space of all functions on $[0, 1]$
• The set of all vectors in $\mathbb{R}^3$ with positive first coordinate is not a subspace of $\mathbb{R}^3$ because it does not contain the zero vector
• The set of all invertible $n \times n$ matrices is not a subspace of the vector space of all $n \times n$ matrices because it is not closed under scalar multiplication (multiplying an invertible matrix by zero yields a non-invertible matrix)

Span of Vectors

Definition and Properties

• The span of a set of vectors $\{v_1, v_2, \ldots, v_k\}$ in a vector space $V$ is the set of all linear combinations of these vectors:
  • $\text{span}(\{v_1, v_2, \ldots, v_k\}) = \{a_1v_1 + a_2v_2 + \ldots + a_kv_k \mid a_1, a_2, \ldots, a_k \in \mathbb{F}\}$, where $\mathbb{F}$ is the field of scalars (e.g., $\mathbb{R}$ or $\mathbb{C}$)
• The span of a set of vectors is always a subspace of the vector space $V$
• If the span of a set of vectors is the entire vector space $V$, then the set is said to span $V$
• The span of the empty set is defined to be the zero vector

Examples of Span

• The span of a single nonzero vector $v$ is the line through the origin in the direction of $v$ (the set of all scalar multiples of $v$)
• The span of two linearly independent vectors in $\mathbb{R}^2$ or $\mathbb{R}^3$ is a plane through the origin
• The span of three linearly independent vectors in $\mathbb{R}^3$ is the entire space
• The span of the vectors $\{(1, 0), (0, 1)\}$ in $\mathbb{R}^2$ is the entire plane $\mathbb{R}^2$

Determining Span

Finding the Span

• To find the span of a set of vectors, one can express an arbitrary vector in the span as a linear combination of the given vectors and then determine the conditions on the coefficients
  • For example, to find the span of $\{(1, 2), (3, 4)\}$ in $\mathbb{R}^2$, let $(x, y)$ be an arbitrary vector in the span:
    • $(x, y) = a(1, 2) + b(3, 4) = (a + 3b, 2a + 4b)$
    • The span is the set of all vectors $(x, y)$ that satisfy the equations $x = a + 3b$ and $y = 2a + 4b$ for some scalars $a$ and $b$

Spanning Sets and Linear Combinations

• If a set of vectors $\{v_1, v_2, \ldots, v_k\}$ spans a vector space $V$, then any vector in $V$ can be expressed as a linear combination of these vectors, although this representation may not be unique
  • For example, the vectors $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ span $\mathbb{R}^3$, so any vector $(x, y, z)$ in $\mathbb{R}^3$ can be written as a linear combination:
    • $(x, y, z) = x(1, 0, 0) + y(0, 1, 0) + z(0, 0, 1)$
• If a vector space $V$ has dimension $n$, then any set of $n$ linearly independent vectors in $V$ spans $V$