Week7-10

Week7-10 - Chapter 3: Vector Spaces March 16, 2009 Lecture...

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Unformatted text preview: Chapter 3: Vector Spaces March 16, 2009 Lecture 16 1 Vector spaces A vector space is a non-empty set V of objects, called vectors , on which are defined two opera- tions, called addition and scalar multiplication : for any vectors u , v in V , the sum u + v is in V ; for a vector u in V and a scalar c (real number), the scalar multiple c u is in V ; subject to the following axioms listed below. The axioms must hold for all vectors u , v , w in V and scalars c and d . 1. u + v = v + u , 2. ( u + v ) + w = u + ( v + w ), 3. There is a vector such that u + = u , 4. For any vector u there is a vector- u such that u + (- u ) = ; 5. c ( u + v ) = c u + c v , 6. ( c + d ) u = c u + d u , 7. c ( d u ) = ( cd ) u , 8. 1 u = u . The vector is called the zero vector of V . The vector- u is called the negative vector of u . By definition of vector space it is easy to see that for any vector u and scalar c , u = , c = ,- u = (- 1) u . For instance, u (3) = 0 u + (4) = 0 u + (0 u + (- u )) (2) = (0 u + 0 u ) + (- u ) (6) = (0 + 0) u + (- u ) = 0 u + (- u ) (4) = ; c = c (0 u ) (7) = ( c 0) u = 0 u = ;- u =- u + =- u + (1- 1) u =- u + u + (- 1) u = + (- 1) u = (- 1) u . Example 1.1. (a) The Euclidean space R n is a vector space under the ordinary addition and scalar multiplication. 1 (b) The set P n of all polynomials of degree less than or equal to n is a vector space under the ordinary addition and scalar multiplication of polynomials. (c) The set M ( m,n ) of all m × n matrices is a vector space under the ordinary addition and scalar multiplication of matrices. (d) The set C [ a,b ] of all continuous functions on the closed interval [ a,b ] is a vector space under the ordinary addition and scalar multiplication of functions. (We do not study this kind of spaces here.) Definition 1.1. Let V and W be vector spaces and W ⊆ V . If the addition and scalar multiplica- tion in W are the same as the addition and scalar multiplication in V , then W is called a subspace of V . If H is a subspace of V , then H is closed for the addition and scalar multiplication of V : if u , v ∈ H and scalar c ∈ R , then u + v ∈ H, c v ∈ H. Theorem 1.2. Let H be a non-empty subset of a vector space V . Then H is a subspace of V if and only if H is closed under addition and scalar multiplication, i.e., (a) For any vectors u , v ∈ H , we have u + v ∈ H , (b) For any scalar c and a vector v ∈ H , we have c v ∈ H . Example 1.2. (a) For a vector space V , the set { } of the zero vector is a subspace, called the zero subspace of V . The whole space V is a subspace of V . The subspaces { } and V are called the trivial subspaces of V . (b) For an m × n matrix A , the set of solutions of the linear system A x = is a subspace of R n ....
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Week7-10 - Chapter 3: Vector Spaces March 16, 2009 Lecture...

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