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Chapter5 - Chapter 5 Vector Subspaces Keith E Emmert Column...

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Chapter 5: Vector Subspaces Keith E. Emmert Column, Row, and Null Spaces Bases and Dimension Coordinate Systems Chapter 5: Vector Subspaces Keith E. Emmert Tarleton State University February 22, 2011
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Chapter 5: Vector Subspaces Keith E. Emmert Column, Row, and Null Spaces Bases and Dimension Coordinate Systems Overview Column, Row, and Null Spaces Bases and Dimension Coordinate Systems
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Chapter 5: Vector Subspaces Keith E. Emmert Column, Row, and Null Spaces Bases and Dimension Coordinate Systems Subsets of Vector Spaces Theorem 1 Let V be a vector space. Suppose U V is nonempty. If U is closed under vector addition and scalar multiplication, then U is a vector space. Proof: We need to check a lot of properties: Additive Properties 1. Closure for Addition If u , v V , then u v V . (True by assumption) 2. Commutativity of Addition If u , v V , then u v = v u . 3. Associativity of Addition If u , v , w V , then ( u v ) w = u ( v w ) . 4. Existence of a Zero Vector There exists 0 V such that for all u V , u 0 = u . 5. Existence of Additive Inverses For all u V , there exists at least one element ˜ u V such that ˜ u u = 0 . Fact: ˜ u is unique! Let ˜ u = - u .
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Chapter 5: Vector Subspaces Keith E. Emmert Column, Row, and Null Spaces Bases and Dimension Coordinate Systems Subsets of Vector Spaces Continued Product Properties 1. Closure for Scalar–Vector Product If α F and u V , then α u V . (True by Assumption) 2. Distributive Law: Scalar Times Vectors For any α F and u , v V , we have α ( u v ) = ( α u ) ( α v ) . 3. Distributive Law: Scalar Sum Times Vector For any α, β F and u V , we have ( α β ) u = ( α u ) ( β u ) . 4. Associativity of Scalar–Vector Product If α, β F and u V , then α ( β u ) = ( α β ) u . 5. Multiplicative Identity: Unit Scalar Times Vector There exists 1 F such that for each u V , 1 u = u .
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Chapter 5: Vector Subspaces Keith E. Emmert Column, Row, and Null Spaces Bases and Dimension Coordinate Systems Subspace Defined Definition 2 A subset in a vector space is a subspace if it is nonempty and closed under the operations of adding vectors and multiplying vectors by scalars. Remark 3 I Since subspaces are closed under scalar multiplication, then 0 v = 0 is in every subspace! I There are two trivial examples of subspaces of a vector space V . Namely, { 0 } and V !
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Chapter 5: Vector Subspaces Keith E. Emmert Column, Row, and Null Spaces Bases and Dimension Coordinate Systems Span of a Nonempty Set Theorem 4 Let S 6 = be a subset of a vector space V . Then Span ( S ) is a subspace. Proof:
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Chapter 5: Vector Subspaces Keith E. Emmert Column, Row, and Null Spaces Bases and Dimension Coordinate Systems Linear Transformations - Kernel and Image Definition 5 If f is a mapping of a set X to a set Y and U X , then the image of U is f [ U ] = { f ( x ) | x U } .
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