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**Unformatted text preview: **Let V be a vector space with operations and and and let W be a non-empty set of vectors of V. If W is a vector space with respect to the operations and in V, then W is called a subspace of V. Sec 4.3 Saturday, February 06, 2010 11:23 PM Section4dot3 Page 1 Let V be a vector space with operations and and and let W be a non-empty set of vectors of V. If W is a vector space with respect to the operations and in V, then W is called a subspace of V. A subspace can be thought of as a vector space W that is a part of a larger vector space V 4.3.1 Saturday, February 06, 2010 11:23 PM Section4dot3 Page 2 Let V be a vector space with operations and and and let W be a non-empty set of vectors of V. If W is a vector space with respect to the operations and in V, then W is called a subspace of V. A subspace can be thought of as a vector space W that is a part of a larger vector space V The elements in W are already vectors since they come from the known vector space V. And W inherits the vector space operations and from V. As a result, to check whether a set W is a vector space, it suffices to check only the closure properties. 4.3.2 Saturday, February 06, 2010 11:23 PM Section4dot3 Page 3 Let V be a vector space with operations and and and let W be a non-empty set of vectors of V. If W is a vector space with respect to the operations and in V, then W is called a subspace of V. A subspace can be thought of as a vector space W that is a part of a larger vector space V The elements in W are already vectors since they come from the known vector space V. And W inherits the vector space operations and from V. As a result, to check whether a set W is a vector space, it suffices to check only the closure properties. If W is closed under and , W inherits the algebraic and inverse properties of V, so all other 8 properties of a vector space are satisfied. 4.3.3 Saturday, February 06, 2010 11:23 PM Section4dot3 Page 4 Let V be a vector space with operations and and and let W be a non-empty set of vectors of V. If W is a vector space with respect to the operations and in V, then W is called a subspace of V. A subspace can be thought of as a vector space W that is a part of a larger vector space V The elements in W are already vectors since they come from the known vector space V. And W inherits the vector space operations and from V. As a result, to check whether a set W is a vector space, it suffices to check only the closure properties. If W is closed under and , W inherits the algebraic and inverse properties of V, so all other 8 properties of a vector space are satisfied. To show that W is a subspace of V, show that a) W is closed under b) W is closed under 4.3.4 (*) Saturday, February 06, 2010 11:23 PM Section4dot3 Page 5 Proof: Let w be any vector in the subspace W. Since W is closed under scalar multiplication , W must contain the vector 0 w = 0, the zero vector in the original ( big ) vector space. DONE 4.3.5 Monday, February 15, 2010 10:20 PM Section4dot3 Page 6 Proof:...

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