Vector Space Definition, Axioms and Examples – Linear Algebra Regulation 2025

Vector Space – Definition, Axioms, Examples (Regulation 2025 Linear Algebra)

Vector Space is one of the most important concepts in Linear Algebra under Regulation 2025 Engineering Mathematics syllabus. This topic forms the foundation for Eigen Values and Eigen Vectors, Subspace, and Basis & Dimension.

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🔷 What is a Vector Space?

A Vector Space (also called Linear Space) is a non-empty set \( V \) together with two operations:

• Vector Addition: \( u + v \)
• Scalar Multiplication: \( c v \)

where \( u, v \in V \) and \( c \in \mathbb{R} \). If certain axioms are satisfied, then \( V \) is called a Vector Space over \( \mathbb{R} \).

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🔷 Formal Definition

Let \( V \) be a non-empty set. If the following axioms hold for all \( u, v, w \in V \) and scalars \( a, b \in \mathbb{R} \), then \( V \) is called a Vector Space.

Vector Space Axioms

1. Closure under Addition: \( u + v \in V \)
2. Commutative Law: \( u + v = v + u \)
3. Associative Law: \( u + (v + w) = (u + v) + w \)
4. Additive Identity: There exists \( 0 \in V \) such that \( u + 0 = u \)
5. Additive Inverse: For each \( u \in V \), there exists \( -u \) such that \( u + (-u) = 0 \)
6. Closure under Scalar Multiplication: \( a u \in V \)
7. Distributive Law: \( a(u + v) = au + av \)
8. Scalar Addition Law: \( (a + b)u = au + bu \)
9. Associativity of Scalars: \( a(bu) = (ab)u \)
10. Identity Scalar: \( 1u = u \)

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🔷 Example: \( \mathbb{R}^2 \) is a Vector Space

Let \( V = \mathbb{R}^2 = \{ (x, y) \mid x, y \in \mathbb{R} \} \)

Addition:

\[ (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2) \]

Scalar Multiplication:

\[ c(x, y) = (cx, cy) \]

Since all 10 axioms are satisfied, \( \mathbb{R}^2 \) is a Vector Space.

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🔷 Example of NOT a Vector Space

Let \( V = \{ (x, y) \mid x, y > 0 \} \)

If scalar \( c = -1 \),

\[ -1(x, y) = (-x, -y) \]

This is not in \( V \). So closure property fails. Hence, it is NOT a Vector Space.

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🔷 Common Exam Mistakes

• Not checking closure property
• Forgetting additive identity
• Ignoring scalar multiplication condition
• Writing incomplete axioms

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🔷 Related Topics (Read Next)

• Subspace – Definition and Examples
• Linear Independence Explained
• Basis and Dimension
• Eigen Values and Eigen Vectors

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🔷 Conclusion

Vector Space is the backbone of Linear Algebra in Regulation 2025 syllabus. Mastering this topic helps in understanding advanced topics like Eigen Values, Inner Product Spaces and Diagonalization.