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vector_spaces1

vector_spaces1 - VECTOR SPACES Many concepts concerning...

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VECTOR SPACES Many concepts concerning vectors in R n can be extended to other mathematical systems. We can think of a vector space in general, as a collection of objects that behave as vectors do in R n . The objects of such a set are called vectors . 1. Vector Space 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 below. The axioms must hold for all u , v and w in V and for all scalars k and m . 1. u + v is in V . 2. u + v = v + u . 3. ( u + v ) + w = u + ( v + w ) 4. There is a vector (called the zero vector) 0 in V such that u + 0 = u . 5. For each u in V , there is vector - u in V satisfying u + ( - u ) = 0 . 6. k u is in V . 7. k ( u + v ) = k u + k v . 8. ( k + m ) u = k u + m u . 9. ( km ) u = k ( m u ) . 10. 1 u = u . 1
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Example The Zero Vector Space Let V consist of a single object, which denote by 0 , and define: 0 +0 = 0 and k 0 +0 = 0 for all skalar k . It is easy to check that all vector space axioms are satisfied. This is called the zero vector space. Some properties of Vectors Let V be a vector space, u a vector in V , and k a scalar, then: (a) 0 u = 0 (b) k 0 = 0 (c) ( - 1) u = -u (d) If k u = 0 , then k = 0 or u = 0 2
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2. Subspaces Vector spaces may be formed from subsets of other vectors spaces.
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