4.2 - MATH1850U/2050U: Chapter 4 cont. GENERAL VECTOR...

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MATH1850U/2050U: Chapter 4 cont. .. 1 GENERAL VECTOR SPACES cont. .. Subspaces (4.2; pg. 179) Definition: A subset W of V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V . Example: Let W consist of the points on the line 1 x y and so, W is a subset of R 2 . Show that W is NOT a subspace of R 2 . Example: The set of points W that lie on a line through the origin is a subset of R 3 . Show that W is a subspace of R 3 . Theorem (Subspaces): If W is a set of one or more vectors from a vector space V , then W is a subspace of V if and only if the following conditions hold: a) If u and v are vectors in W , then v u is in W b) If k is any scalar and u is any vector in W , then u k is in W
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MATH1850U/2050U: Chapter 4 cont. .. 2 Example: The set of all points on a plane is a subset of R 3 . Every plane through the origin is a subspace of R 3 . Example: In the previous section, we saw that the set that contained only the zero vector was a vector space (called the zero vector space). Because every vector space has a zero vector, the
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4.2 - MATH1850U/2050U: Chapter 4 cont. GENERAL VECTOR...

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