MA25C02 Unit 1: Vector Subspaces & Solved Examples (Regulation 2025) 🎓🔢

▶ Tutorial Video: Vector Spaces-Subspaces-Defintion

Vector Subspaces

Definition: Subspace A subset \(W\) of a vector space \(V\) over a field \(F\) is called a subspace if \(W\) is itself a vector space under the same operations defined on \(V\).

---

Necessary Properties:

• \(x + y \in W\) whenever \(x, y \in W\) (Additive Closure)
• \(\alpha x \in W\) whenever \(\alpha \in F, x \in W\) (Scalar Closure)
• \(W\) contains the zero vector \(\mathbf{0}\)
• Every vector in \(W\) has an additive inverse in \(W\)

Linear & Direct Sums

Linear Sum (\(W_1 + W_2\)) \[ W_1 + W_2 = \{ v \in V \mid v = w_1 + w_2, w_1 \in W_1, w_2 \in W_2 \} \]

Direct Sum (\(V = W_1 \oplus W_2\)) \(V\) is the direct sum of \(W_1\) and \(W_2\) if:

1. \(V = W_1 + W_2\)
2. \(W_1 \cap W_2 = \{ \mathbf{0} \}\)

Worked Examples

Question 1 A/M-25-R-21

Is the zero vector a subspace? Give reason.

Yes, the zero vector alone forms a subspace.

Reason:

Let \(W = \{ \mathbf{0} \}\):

• \(W\) is non-empty (\(\mathbf{0} \in W\)).
• Closed under Addition: \(\mathbf{0} + \mathbf{0} = \mathbf{0} \in W\).
• Closed under Scalar: \(c \cdot \mathbf{0} = \mathbf{0} \in W\).

Question 2 N/D-24-R-21

Determine whether the set of vectors of the form \((a, b, 1)\) is a subspace of \(\mathbb{R}^3\).

Let \(W = \{ (a, b, 1) \mid a, b \in \mathbb{R} \}\)

Verification: To be a subspace, \(W\) must contain the zero vector \((0,0,0)\).

• In \(\mathbb{R}^3\), the zero vector is \((0,0,0)\).
• However, every vector in \(W\) has a fixed third component of 1.
• Therefore, \((0,0,0) \notin W\).

Conclusion: W is NOT a subspace of \(\mathbb{R}^3\).