la03-Vector_Spaces_Part1

la03-Vector_Spaces_Part1 - Chapter 3 Vector Spaces Chapter...

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Chapter 3: Vector Spaces
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2 Chapter 3: Overview Definition and Examples Subspaces Linear Independence Basis and Dimension Change of Basis Row Space and Column Space
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Vector Spaces Vector spaces or linear spaces Operations satisfy closure property Addition Scalar multiplication A set of elements on which the operations of addition and scalar multiplication are defined. The set together with the operations (addition + scalar multiplication) form a vector space if a set of axioms are satisfied. 3
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Euclidean Vector Spaces, R n R 2 A nonzero vector Euclidean length of x is 4
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Euclidean Vector Spaces, R n R 2 For each scalar α , The sum of two vectors u and v is defined by 5
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Euclidean Vector Spaces, R n R 3 R n For any x , y in R n and any scalar α , 6
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Vector Spaces R m×n Let R m×n denote the set of all m×n matrices with real entries. If A =( a ij ) and B =( b ij ), α is a scalar, α A = C =( c ij ), where c ij = α a ij the sum A + B is defined to be A + B = C =( c ij ), where c ij = a ij + b ij 7
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Vector Space Axioms Let V be a set of elements, called vectors , on which the operations of addition and scalar multiplication are defined. Closure properties: 8
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Vector Space Axioms The set V together with the operations of addition and scalar multiplication is said to form a vector space if the following axioms are satisfied: 9
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To check if a set V together with addition and scalar multiplication form a vector space we need to check: Satisfy the closure properties C1 and C2. Satisfy eight axioms. Examples
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la03-Vector_Spaces_Part1 - Chapter 3 Vector Spaces Chapter...

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