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MATH2107-4-11

# MATH2107-4-11 - u ∈ H and for each scalar c c u ∈ H —...

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Section 4.1 Vector Spaces and Subspaces Linear Algebra and its Applications David C. Lay — Winter 2008 – p. 1/10

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Vector Spaces and Subspaces Recall in § 1 . 3 we looked at vector equations where our vectors were from R 2 , R 3 , R 4 ,..., R n ,n 1 . R 2 , R 3 , R n are all examples of vector spaces . A vector space is a non-empty set of elements called vectors . The set is usually denoted by V and an element in V by v . v V v is an element of V Two operations are defined on V , called addition and multiplication by scalars subject to the following axioms. — Winter 2008 – p. 2/10
For all u , v , w V and scalars c,d 1. u + v V 2. u + v = v + u 3. ( u + v ) + w = u + ( v + w ) 4. There is a zero vector 0 V such that u + 0 = u . 5. For each u V there is a vector - u V such that u + ( - u ) = 0 6. c u V 7. c ( u + v ) = c u + c v 8. ( c + d ) u = c u + c v 9. c ( d u ) = ( cd ) u 10. 1 u = u — Winter 2008 – p. 3/10

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Subspaces A subsect H of a vector space V satisfying the following three properties is called a subspace 1. 0 V is also in H . i.e. 0 H . 2. H is closed under addition. i.e. for all u , v H, u + v H 3. H is closed under scalar multiplication. i.e. for all

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Unformatted text preview: u ∈ H and for each scalar c, c u ∈ H . — Winter 2008 – p. 4/10 Examples For any u ∈ V and every scalar c 1. u = 2. c = 3.-u = (-1) u where is the zero vector. — Winter 2008 – p. 5/10 Example Let H = { } be a subset of V . Is H a subspace of V ? — Winter 2008 – p. 6/10 Example Is R 2 a subspace of R 3 ? — Winter 2008 – p. 7/10 Example Is H = s t ; s, t ∈ R a subspace of R 3 ? — Winter 2008 – p. 8/10 Example Let H = span { v 1 , v 2 } where v 1 , v 2 ∈ V . Show H is a subspace of V . — Winter 2008 – p. 9/10 Example H = 2 t-t ; t ∈ R . Show H is a subspace of R 3 . — Winter 2008 – p. 10/10...
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MATH2107-4-11 - u ∈ H and for each scalar c c u ∈ H —...

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