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
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
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
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
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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