Linear Algebra: UNIT I: Vector Spaces

Vector Spaces: Theorems Part 5

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Important Theorems for Engineering Maths or Mathematics - Vector Spaces: Theorems Part 5

Vector Spaces

Theorems Part 5

Theorem 14

Let V be a vector space over F and W be a subspace of V then L(W) = W.

Proof:

Let W V

Then L(W) W. ………(1)

To prove: W L(W)

Let v ∈ W

lv ∈ L (W) → Linear combinations

v ∈ L (W)

W L(W). ………(2)

From (1) and (2), L (W) = W.

Note

1. Let V be a vector space over F and W is contained in V and L (W) = W, then W is a sub space of V.

2. Let V be a vector space over F and S₁ V, S₂ V then L(S₁ ∪ S₂)=L(S₁) + L (S₂).

Linear Algebra: UNIT I: Vector Spaces : Tag: maths, mathematics : - Vector Spaces: Theorems Part 5

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Linear Algebra: UNIT I: Vector Spaces

Introduction of Vector Spaces

Definition of Vector Spaces

Vector Spaces: Basic Example Problems

Vector Spaces: Theorems Part 1

Vector Spaces: Theorems Part 1: Example Solved Problems

Definition and Theorem of Subspaces

Subspaces (Vector Spaces): Example Solved Problems

Vector Subspaces: Theorems Part 2

Vector SubSpaces: Theorems Part 2 - Example Solved Problems

Linear Combination

Linear Combination: Example Solved Problems

Linear span

Linear span: Example Solved Problems

Vector Spaces: Theorems Part 3

Vector Spaces: Theorems Part 3 - Example Solved Problems

Vector Spaces: Theorems Part 4

Vector Spaces: Theorems Part 4 - Example Solved Problems

Vector Spaces: Theorems Part 5

Definition of Linear Dependence and Linear Independence

Linear Dependence and Linear Independence: Example Solved Problems

Vector Spaces: Theorems Part 6

Vector Spaces: Theorems Part 6 - Example Solved Problems

Bases and Dimensions

Bases and Dimensions: Example Solved Problems

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