Span and Intersection of Subspaces

Author: Bindeshwar Singh Kushwaha
Institute: PostNetwork Academy

Linear Span: Definition and Properties

• Given vectors \( u_1, u_2, \ldots, u_m \) in a vector space \( V \), their linear span is the set of all linear combinations:
  \[
  \text{span}(u_1, u_2, \ldots, u_m) = \{ a_1 u_1 + a_2 u_2 + \cdots + a_m u_m \mid a_i \in \mathbb{K} \}
  \]
• The zero vector always belongs to the span.
• The span is closed under vector addition and scalar multiplication.
• The span of a set of vectors is a subspace of \( V \).
• Theorem:
  • (i) \( \text{span}(S) \) is a subspace of \( V \) that contains \( S \).
  • (ii) If \( W \) is a subspace of \( V \) containing \( S \), then \( \text{span}(S) \subseteq W \).

Numerical Examples of Linear Spans

• Example 1: Span of \( u_1 = (1,0) \) and \( u_2 = (0,1) \) in \( \mathbb{R}^2 \) is the entire plane \( \mathbb{R}^2 \).
• Example 2: Span of \( u_1 = (1,2) \) and \( u_2 = (2,4) \) is a line in \( \mathbb{R}^2 \) since \( u_2 \) is a multiple of \( u_1 \).
• Example 3: Span of \( u_1 = (1,0,0) \), \( u_2 = (0,1,0) \) in \( \mathbb{R}^3 \) is the \( xy \)-plane.
• Example 4: Span of \( u_1 = (1,1,1) \), \( u_2 = (0,1,2) \), \( u_3 = (1,0,1) \) in \( \mathbb{R}^3 \) is the entire space \( \mathbb{R}^3 \).
• Example 5: Span of the rows of matrix \( A = \begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix} \) is the line defined by \( (1,2) \) because the second row is a multiple of the first.

Intersection of Subspaces – Explanation

• Let \( U \) and \( W \) be subspaces of a vector space \( V \).
• The zero vector belongs to both \( U \) and \( W \), so \( 0 \in U \cap W \).
• Let \( u, v \in U \cap W \), so \( u, v \in U \) and \( u, v \in W \).
• Since \( U \) and \( W \) are subspaces, \( au + bv \in U \) and \( au + bv \in W \) for all scalars \( a, b \in K \).
• Therefore, \( au + bv \in U \cap W \).
• Hence, \( U \cap W \) is a subspace of \( V \).

Theorem

Theorem: The intersection of any number of subspaces of a vector space \( V \) is a subspace of \( V \).

Numerical Example 1

• In \( \mathbb{R}^2 \): \( U = \text{span}\{(1,0)\} \), \( W = \text{span}\{(0,1)\} \)
• Intersection: \( U \cap W = \{(0, 0)\} \)

Numerical Example 2

• In \( \mathbb{R}^3 \): \( U = \text{span}\{(1,0,0), (0,1,0)\} \)
• \( W = \text{span}\{(1,1,0)\} \)
• Intersection: \( U \cap W = \text{span}\{(1,1,0)\} \)

Numerical Example 3

• \( A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{bmatrix} \)
• \( B = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \)
• Assume \( B \) lies in the column space of \( A \), so the intersection is: \( \text{span}\{(1, 2, 3)^T\} \)

Numerical Example 4

• \( U = \text{span}\{(1, 2, 3), (4, 5, 6)\} \)
• \( W = \text{span}\{(1, 0, 0), (0, 1, 0)\} \)
• The intersection lies in the XY-plane; the exact intersection can be found by solving the system.

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