Lect11 - Chapter 3 Vector Spaces Math1111 Subspaces...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations. Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations. Proof. • Addition is well-defined in S . Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations. Proof. • Addition is well-defined in S . Let ( a b ) T , ( x y ) T ∈ S . Then a = 2 b and x = 2 y . Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations. Proof. • Addition is well-defined in S . Let ( a b ) T , ( x y ) T ∈ S . Then a = 2 b and x = 2 y . Thus ( a b ) T +( x y ) T = ( a + x b + y ) T and a + x = 2 b + 2 y = 2 ( b + y ) . Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations. Proof. • Addition is well-defined in S . Let ( a b ) T , ( x y ) T ∈ S . Then a = 2 b and x = 2 y . Thus ( a b ) T +( x y ) T = ( a + x b + y ) T and a + x = 2 b + 2 y = 2 ( b + y ) . Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations. Proof. • Addition is well-defined in S . Let ( a b ) T , ( x y ) T ∈ S . Then a = 2 b and x = 2 y . Thus ( a b ) T +( x y ) T = ( a + x b + y ) T and a + x = 2 b + 2 y = 2 ( b + y ) . ∴ ( a + x b + y ) T ∈ S Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations....
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This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.

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Lect11 - Chapter 3 Vector Spaces Math1111 Subspaces...

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