![]() ![]() ![]() The set consisting of only the zero vector in a vector space V is a subspace of V, called the zero subspace and written as in H such that. ![]() Conversely, every vector space is a subspace (of itself and possibly of other larger spaces). Properties (a), (b), and (c) guarantee that a subspace H of V is itself a vector space, under the vector space operations already defined in V. That is, for each u in H and each scalar c, the vector cu is in H. © 2016 Pearson Education, Inc.ħ SUBSPACES H is closed under multiplication by scalars. That is, for each u and v in H, the sum is in H. © 2016 Pearson Education, Inc.Ħ SUBSPACES Definition: A subspace of a vector space V is a subset H of V that has three properties: The zero vector of V is in H. So Axioms 1, 4, 5, 6, and 10 are evident. An arrow of zero length is a single point and represents the zero vector. Solution: The definition of V is geometric, using concepts of length and direction. Define addition by the parallelogram rule, and for each v in V, define cv to be the arrow whose length is times the length of v, pointing in the same direction as v if and otherwise pointing in the opposite direction. © 2016 Pearson Education, Inc.įor each u in V and scalar c, (1) (2) (3) Example 2: Let V be the set of all arrows (directed line segments) in three-dimensional space, with two arrows regarded as equal if they have the same length and point in the same direction. Using these axioms, we can show that the zero vector in Axiom 4 is unique, and the vector, called the negative of u, in Axiom 5 is unique for each u in V. There is a zero vector 0 in V such that © 2016 Pearson Education, Inc.įor each u in V, there is a vector in V such that The scalar multiple of u by c, denoted by cu, is in V. The sum of u and v, denoted by, is in V. The axioms must hold for all vectors u, v, and w in V and for all scalars c and d. A nonempty subset S of V is referred to as a subspace of V, if and only if it satisfies all the following. Presentation on theme: "VECTOR SPACES AND SUBSPACES"- Presentation transcript:ĭefinition: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the ten axioms (or rules) listed below. Suppose V is a vector space over the field F. ![]()
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