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 }⊂ 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 ∈ S where is the zero vector in S . (A4) - require- x ∈ S whenever x ∈ S . 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 ∈ S where is the zero vector in S . (A4) - require- x ∈ S whenever x ∈ S . Both OK if α x ∈ S 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 . 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. 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...
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Lect11 - Chapter 3. Vector Spaces Math1111 Subspaces...

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