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Lect11 - Chapter 3 Vector Spaces Math1111 Subspaces...

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Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } ⊂ R 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. Question Given a vector space W , is every nonempty subset a vector space w.r.t. the operations of W ? Ans. NO Observation Inherited from V , elements in S fulfill (A1), (A2), (A5)-(A8). But, x + y ? S for x , y S and α x ? S for x S and scalar α . (A3) - require 0 S where 0 is the zero vector in S . (A4) - require - x S whenever x S .
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Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } ⊂ R 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. Question Given a vector space W , is every nonempty subset a vector space w.r.t. the operations of W ? Ans. NO Observation Inherited from V , elements in S fulfill (A1), (A2), (A5)-(A8). But, x + y ? S for x , y S and α x ? S for x S and scalar α . (A3) - require 0 S where 0 is the zero vector in S . (A4) - require - x S whenever x S . Both OK if α x S
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Chapter 3. Vector Spaces Math1111 Subspaces Definition & Examples Definition Let ; 6 = S V where V is a vector space. If (i) α x S for any scalar α and any x S , and (ii) x + y S whenever x , y S , then S is said to be a subspace of V .
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Chapter 3. Vector Spaces Math1111 Subspaces Definition & Examples Definition Let ; 6 = S V where V is a vector space. If (i) α x S for any scalar α and any x S , and (ii) x + y S whenever x , y S , then S is said to be a subspace of V . Example . Show that S = ( x 1 x 2 ) T : x 1 = x 2 is a subspace of R 2 . Describe S geometrically.
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Chapter 3. Vector Spaces Math1111 Subspaces Definition & Examples Definition Let ; 6 = S V where V is a vector space. If (i) α x S for any scalar α and any x S , and (ii) x + y S whenever x , y S , then S is said to be a subspace of V .
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