Week7_moodle_2019d2d4.pdf - MA 106-Week 7 Linear Algebra Neela Nataraj Department of Mathematics Indian Institute of Technology Bombay Powai Mumbai 76

Week7_moodle_2019d2d4.pdf - MA 106-Week 7 Linear Algebra...

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MA 106 -Week 7: Linear Algebra Neela Nataraj Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 76 [email protected] February 19, 2019 Neela Nataraj Week 7
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Outline of the lecture Vector Spaces Linear transformations Inner Product Spaces Neela Nataraj Week 7
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Vector Spaces A vector space over a scalar field F ( R or C in this course) is a triple ( V , + , · ) , where V is a non-empty set, + denotes vector addition and · denotes scalar multiplication satifying the properties: for x , y , z V , α, β F , 1. x + y V Closed w.r.t. addition 2. α · v V Closed w.r.t. scalar mult. 3. x + y = y + x Commutativity of addition 4. ( x + y ) + z = x + ( y + z ) Associativity of addition 5. There exists a unique zero vector denoted 0 V satisfying x + 0 = x Existence of Additive identity 6. For every vector x , there is a unique associated vector - x such that x + - x = 0 Existence of Additive Inverse 7. 1 · x = x Rule of multiplication by 1 8. α · ( β x ) = ( αβ ) · x Associativity of multiplication by scalars 9. ( α + β ) · x = α · x + β · x Distributivity of scalar mult. over field addition 10. α · ( x + y ) = α · x + α · y Distributivity of scalar mult. over vector addition Neela Nataraj Week 7
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Recall: Subspace of a vector space A subspace of V is any subset that satisfies the two requirements : 1 If we add any two vectors x and y in the subspace, their sum x + y is in the subspace. 2 If we multiply any vector x in the subspace by any scalar α , the vector α · x is still in the subspace. That is, a subspace is a subset of a vector space which is closed under addition and scalar multiplication. There is no need to verify the eight required properties, as they are satisfied in the larger space and will be automatically satisfied in every subspace. In particular, zero vector will belong to every subspace (choose scalar to be 0). Neela Nataraj Week 7
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Recall: Linear combinations Definition (Linear combinations) Given v 1 , v 2 , ..., v k V a linear combination is a vector c 1 v 1 + c 2 v 2 + · · · + c k v k for any choice of scalars c 1 , c 2 , ..., c k . Definition (Linear dependence) A set of vectors v 1 , v 2 , ..., v k V is called linearly dependent If scalars c 1 , c 2 , ..., c k , at least one NON-zero, can be found such that c 1 v 1 + c 2 v 2 + · · · + c k v k = 0 . Contrapositive of linear dependence is linear independence. Neela Nataraj Week 7
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Recall: Linear span, basis Definition (Linear span) Given a (finite) set S of vectors in a vector space V , its linear span L ( S ) is the set of ALL linear combinations of elements of S . Definition (Basis) If a set of vectors S in a vector space V is such that S is linearly independent and Every vector in V is a (finite) linear combination of vectors from S . then the set S is called a basis of S . Definition (Dimension) The cardinality of any basis of V is called the dimension of V .
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