# Lectures-8-9.pdf - Linear Algebra Department of Mathematics...

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Linear Algebra Department of Mathematics Indian Institute of Technology Guwahati January – May 2019 MA 102 ( RA, RKS, MGPP, KVK ) 1 / 21
Vector spaces Topics: Vector Spaces and Subspaces Linear Independence Basis and Dimension 2 / 21
Field axioms A field is a set F with two binary operations called addition, denoted by +, and multiplication, denoted by · , satisfying the following field axioms: 3 / 21
Field axioms A field is a set F with two binary operations called addition, denoted by +, and multiplication, denoted by · , satisfying the following field axioms: 1 Closure: For all α, β F , the sum α + β F and the product α · β F . 3 / 21
Field axioms A field is a set F with two binary operations called addition, denoted by +, and multiplication, denoted by · , satisfying the following field axioms: 1 Closure: For all α, β F , the sum α + β F and the product α · β F . 2 Commutativity: For all α, β F , α + β = β + α and α · β = β · α . 3 / 21
Field axioms A field is a set F with two binary operations called addition, denoted by +, and multiplication, denoted by · , satisfying the following field axioms: 1 Closure: For all α, β F , the sum α + β F and the product α · β F . 2 Commutativity: For all α, β F , α + β = β + α and α · β = β · α . 3 Associativity: For all α, β, γ, ( α + β ) + γ = α + ( β + γ ) and ( α · β ) · γ = α · ( β · γ ). 3 / 21
Field axioms A field is a set F with two binary operations called addition, denoted by +, and multiplication, denoted by · , satisfying the following field axioms: 1 Closure: For all α, β F , the sum α + β F and the product α · β F . 2 Commutativity: For all α, β F , α + β = β + α and α · β = β · α . 3 Associativity: For all α, β, γ, ( α + β ) + γ = α + ( β + γ ) and ( α · β ) · γ = α · ( β · γ ). 4 Identity: There exist 0 F and 1 F such that α + 0 = α and 1 · α = α for all α F . 3 / 21
Field axioms A field is a set F with two binary operations called addition, denoted by +, and multiplication, denoted by · , satisfying the following field axioms: 1 Closure: For all α, β F , the sum α + β F and the product α · β F . 2 Commutativity: For all α, β F , α + β = β + α and α · β = β · α . 3 Associativity: For all α, β, γ, ( α + β ) + γ = α + ( β + γ ) and ( α · β ) · γ = α · ( β · γ ). 4 Identity: There exist 0 F and 1 F such that α + 0 = α and 1 · α = α for all α F . 5 Inverse: For α F , there exist β, γ F such that α + β = 0, and α · γ = 1 when α 6 = 0 . 3 / 21
Field axioms A field is a set F with two binary operations called addition, denoted by +, and multiplication, denoted by · , satisfying the following field axioms: 1 Closure: For all α, β F , the sum α + β F and the product α · β F . 2 Commutativity: For all α, β F , α + β = β + α and α · β = β · α . 3 Associativity: For all α, β, γ, ( α + β ) + γ = α + ( β + γ ) and ( α · β ) · γ = α · ( β · γ ). 4 Identity: There exist 0 F and 1 F such that α + 0 = α and 1 · α = α for all α F . 5 Inverse: For α F , there exist β, γ F such that α + β = 0, and α · γ = 1 when α 6 = 0 . β is denoted by - α and γ by α - 1 or 1 /α. 3 / 21
Fields axioms (cont.)