Vector Subspace in Linear Algebra

Author: Bindeshwar Singh Kushwaha
Institute: PostNetwork Academy

Definition of a Subspace

Let \( V \) be a vector space over a field \( K \) and let \( W \subseteq V \).
Then \( W \) is a subspace of \( V \) if \( W \) is itself a vector space over \( K \) with respect to vector addition and scalar multiplication from \( V \).

Note: If \( W \subseteq V \) and satisfies the vector space axioms, then it is a subspace. We use a simplified criterion below.

Subspace Criterion

Suppose \( W \subseteq V \). Then \( W \) is a subspace of \( V \) if:

• The zero vector \( \vec{0} \in W \)
• For all \( u, v \in W \), \( u + v \in W \)
• For all \( u \in W \) and \( k \in K \), \( ku \in W \)

Equivalently: For any \( u, v \in W \) and scalars \( a, b \in K \), \( au + bv \in W \).

Trivial Subspaces

• Every vector space \( V \) has two subspaces:
  • \( \{0\} \): the zero subspace
  • \( V \): the whole space
• These are called the trivial subspaces of \( V \).

Example: A Subspace of \( \mathbb{R}^3 \)

Let \( V = \mathbb{R}^3 \), and define:

\[
U = \{ (a, b, c) \in \mathbb{R}^3 \mid a = b = c \}
\]

Then \( U \) contains all vectors like:

\[
(1,1,1), (-3,-3,-3), (7,7,7)
\]

Check Subspace Conditions:

• \( \vec{0} = (0, 0, 0) \in U \)
• Closed under addition: if \( u = (a, a, a), v = (b, b, b) \Rightarrow u + v = (a+b, a+b, a+b) \in U \)
• Closed under scalar multiplication: \( ku = (ka, ka, ka) \in U \)
• Hence, \( U \) is a subspace of \( \mathbb{R}^3 \).

Example: Line Subspace \( U = \{(a, a, a)\} \)

• Let \( U = \{(a,a,a): a \in \mathbb{R}\} \)
• It passes through the origin \( (0,0,0) \)
• Closed under addition: \( (a,a,a)+(b,b,b)=(a+b,a+b,a+b)\in U \)
• Closed under scalar multiplication: \( k(a,a,a)=(ka,ka,ka)\in U \)
• Hence, \( U \) is a subspace of \( \mathbb{R}^3 \)

Example: Plane Subspace \( W \subseteq \mathbb{R}^3 \)

• Let \( W \) be a plane in \( \mathbb{R}^3 \) passing through origin
• \( 0 = (0,0,0) \in W \)
• For \( u, v \in W \), the sum \( u+v \in W \)
• For scalar \( k \), \( ku \in W \)
• Hence, \( W \) is a subspace of \( \mathbb{R}^3 \)

Subspaces of Matrix Spaces

Let \( V = M_{n,n} \), the vector space of \( n \times n \) matrices.

• Let \( W_1 \): subset of all upper triangular matrices
• Let \( W_2 \): subset of all symmetric matrices
• \( W_1 \subseteq V \) because:
  • Contains zero matrix \( 0 \)
  • Closed under matrix addition
  • Closed under scalar multiplication
• So, \( W_1 \) is a subspace of \( V \)
• Similarly, \( W_2 \) is also a subspace of \( V \)

Subspaces of Polynomial Space

Let \( V = P(t) \), the vector space of polynomials.

• Let \( P_n(t) \): polynomials of degree at most \( n \)
• Let \( Q(t) \): polynomials with only even powers of \( t \)
• Then \( Q(t) \subseteq P(t) \)
• Example polynomials in \( Q(t) \):
  \[
  \begin{aligned}
  p_1 &= 3 + 4t^2 – 5t^6 \\
  p_2 &= 6 – 7t^4 + 9t^6 + 3t^{12}
  \end{aligned}
  \]
• Constant \( k = kt^0 \) is an even power → included
• So, \( Q(t) \) is a subspace of \( P(t) \)

Subspaces of Real-Valued Functions

Let \( V \): vector space of real-valued functions.

• Let \( W_1 \): collection of all continuous functions
  • Example: \( f(x) = \sin x \), \( g(x) = |x| \)
• Let \( W_2 \): collection of all differentiable functions
  • Example: \( h(x) = x^2 \), \( p(x) = e^x \)
• Both \( W_1 \) and \( W_2 \) are closed under:
  • Addition: \( f(x) + g(x) \), \( h(x) + p(x) \)
  • Scalar multiplication: \( 3f(x) \), \( -2h(x) \)
• So, \( W_1 \) and \( W_2 \) are subspaces of \( V \)

PDF

Vector Subspace in Linear Algebra

Video

Reach PostNetwork Academy

• Website: www.postnetwork.co
• YouTube Channel: youtube.com/@postnetworkacademy
• Facebook Page: facebook.com/postnetworkacademy
• LinkedIn Page: linkedin.com/company/postnetworkacademy
• GitHub Repositories: github.com/postnetworkacademy

Thank You!