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

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Unformatted text preview: 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 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 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 V such that for all u V , u = u . 5. Existence of Additive Inverses For all u V , there exists at least one element u V such that u u = . Fact: u is unique! Let u =- u . 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 ScalarVector Product If F and u V , then u V . (True by Assumption) 2. Distributive Law: Scalar Times Vectors For any F and u , v V , we have ( u v ) = ( u ) ( v ) . 3. Distributive Law: Scalar Sum Times Vector For any , F and u V , we have ( ) u = ( u ) ( u ) . 4. Associativity of ScalarVector Product If , F and u V , then ( u ) = ( ) u . 5. Multiplicative Identity: Unit Scalar Times Vector There exists 1 F such that for each u V , 1 u = u . 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 v = is in every subspace! I There are two trivial examples of subspaces of a vector space V. Namely, { } and V! 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: 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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This note was uploaded on 01/16/2012 for the course MATH 332 taught by Professor Keithemmert during the Spring '11 term at Tarleton.

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

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