ch4 General Vector Spaces
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Ch04_4
A Vector Space in R3
補充： Let W { a(1, 0, 1) | a R }. Prove that W is a vector space. Proof Let u a(1,0,1) and v b(1,0,1) W , for some a, bR. Axiom 1: u v a(1,0,1) b(1,0,1) (a b)(1,0,1) u + v W. Thus W is closed under addition. Axiom 2: cu ca(1,0,1) W. Thus W is closed under scalar multiplication. Axiom 5: Let 0 = (0, 0, 0) = 0(1,0,1), then 0 W and 0+u = u+0 = u for any u W. Axiom 6: For any u = a(1,0,1) W. Let -u = -a(1,0,1), then -u W and (-u)+u = 0. 隨堂作業：5 Axiom 3,4 and 7~10: trivial
Ch04_11
Subspaces
In general, a subset of a vector space may or may not satisfy the closure axioms. However, any subset that is closed under both of these operations satisfies all the other vector space properties.
Ch04_2
補充範例
(1) V={ …, -3, -1, 1, 3, 5, 7, …} (所有奇數構成的集合) V is not closed under addition because 1+3=4 V.
(2)
பைடு நூலகம்
Z={ …, -2, -1, 0, 1, 2, 3, 4, …} (所有整數構成的集合) Z is closed under addition because for any a, b Z, a + b Z. Z is not closed under scalar multiplication because ½ is a scalar, for any odd a Z, (½)a Z.
Axiom 3 and 4: From our previous discussions we know that 2 2 matrices are communicative and associative under addition (Theorem 2.2).
Ch04_6
Axiom 5: 0 0 The 2 2 zero matrix is 0 0 0 , since a b 0 0 a b u u0 c d 0 0 c d Axiom 6: a b - a - b If u , then - u , since c d - c - d a b - a - b a - a b - b 0 0 0 u ( -u ) c d - c - d c - c d - d 0 0 推廣：The set of m n matrices, Mmn, is a vector space.
Ch04_5
Vector Spaces of Matrices
p q Let M 22 { | p, q, r , s R }. Prove that M22 is a vector space. r s
Proof
a b e f Let u and v g h M 22 . c d Axiom 1: a b e f a e b f uv c d g h c g d h u + v is a 2 2 matrix. Thus M22 is closed under addition.
Linear Algebra
Chapter 4 General Vector Spaces
大葉大學 資訊工程系 黃鈴玲
4.1 General Vector Spaces and Subspaces
Our aim in this section will be to focus on the algebraic properties of Rn. We draw up a set of axioms (公理) based on the properties of Rn. Any set that satisfies these axioms will have similar algebraic properties to the vector space Rn.
Note. U只要有加法與純量乘法的封閉性，其他axiom都會滿足。
(跳過)
Ch04_10
Theorem 4.1
Let V be a vector space, v a vector in V, 0 the zero vector of V, c a scalar, and 0 the zero scalar. Then (a) 0v = 0 (b) c0 = 0 (c) (-1)v = -v (d) If cv = 0, then either c = 0 or v = 0. Proof (a) 0v + 0v = (0 + 0)v = 0v (0v + 0v) + (-0v) = 0v + (-0v) 0v + [0v + (-0v)] = 0, 0v + 0 = 0, 0v = 0 (c) (-1)v + v = (-1)v + 1v = [(-1) + 1]v = 0v = 0
p q 隨堂作業：7 W { | p, q, r , s 0}. r s
Ch04_7
Vector Spaces of Functions
Prove that V={ f | f: R R } is a vector space.
(跳過)
Let f, g V, c R. For example: f: R R, f(x)=2x, g: R R, g(x)=x2+1. Axiom 1: f + g is defined by (f + g)(x) = f(x) + g(x). f + g : R R f + g V. Thus V is closed under addition. Axiom 2: cf is defined by (cf)(x) = c f(x). cf : R R cf V. Thus V is closed under scalar multiplication.
(u1 , ..., un ) (v1 , ..., vn ) (u1 v1 , ..., un vn ) c(u1 , ..., un ) (cu1 , ..., cun )
It can be shown that Cn with these two operations is a complex vector space.
Definition
A vector space is a set V of elements called vectors, having operations of addition and scalar multiplication defined on it that satisfy the following conditions. (u, v, and w are arbitrary elements of V, and c and d are scalars.) Closure Axioms (最重要！) 1. The sum u + v exists and is an element of V. (V is closed under addition.) 2. cu is an element of V. (V is closed under scalar multiplication.)
Ch04_3
隨堂作業：14
Definition of Vector Space (continued)
Addition Axioms 3. u + v = v + u (commutative property) 4. u + (v + w) = (u + v) + w (associative property) 5. There exists an element of V, called the zero vector, denoted 0, such that u + 0 = u. 6. For every element u of V there exists an element called the negative of u, denoted -u, such that u + (-u) = 0. Scalar Multiplication Axioms 7. c(u + v) = cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10.1u = u
Definition
Let V be a vector space and U be a nonempty subset of V. If U is a vector space under the operations of addition and scalar multiplication of V it is called a subspace of V.
Ch04_8
Vector Spaces of Functions
(跳過)
Axiom 5: Let 0 be the function such that 0(x) = 0 for every xR. 0 is called the zero function. We get (f + 0)(x) = f(x) + 0(x) = f(x) + 0 = f(x) for every xR. Thus f + 0 = f. 0 is the zero vector. Axiom 6: Let the function –f defined by (-f )(x) = -f (x).
[ f (- f )]( x) f ( x) (- f )( x) f ( x) - [ f ( x)] 0 0( x )
Thus [f + (-f )] = 0, -f is the negative of f. 隨堂作業：13 V={ f | f(x)=ax2+bx+c for some a,b,c R}
Ch04_9
The Complex Vector Space Cn
Let (u1 , ..., un ) be a sequenceof n complex numbers. The set of all such sequencesis denotedCn . Let operationsof addition and scalar multiplica tion (by a complex scalar c) be defined on Cn as follows :
A Vector Space in R3
補充： Let W { a(1, 0, 1) | a R }. Prove that W is a vector space. Proof Let u a(1,0,1) and v b(1,0,1) W , for some a, bR. Axiom 1: u v a(1,0,1) b(1,0,1) (a b)(1,0,1) u + v W. Thus W is closed under addition. Axiom 2: cu ca(1,0,1) W. Thus W is closed under scalar multiplication. Axiom 5: Let 0 = (0, 0, 0) = 0(1,0,1), then 0 W and 0+u = u+0 = u for any u W. Axiom 6: For any u = a(1,0,1) W. Let -u = -a(1,0,1), then -u W and (-u)+u = 0. 隨堂作業：5 Axiom 3,4 and 7~10: trivial
Ch04_11
Subspaces
In general, a subset of a vector space may or may not satisfy the closure axioms. However, any subset that is closed under both of these operations satisfies all the other vector space properties.
Ch04_2
補充範例
(1) V={ …, -3, -1, 1, 3, 5, 7, …} (所有奇數構成的集合) V is not closed under addition because 1+3=4 V.
(2)
பைடு நூலகம்
Z={ …, -2, -1, 0, 1, 2, 3, 4, …} (所有整數構成的集合) Z is closed under addition because for any a, b Z, a + b Z. Z is not closed under scalar multiplication because ½ is a scalar, for any odd a Z, (½)a Z.
Axiom 3 and 4: From our previous discussions we know that 2 2 matrices are communicative and associative under addition (Theorem 2.2).
Ch04_6
Axiom 5: 0 0 The 2 2 zero matrix is 0 0 0 , since a b 0 0 a b u u0 c d 0 0 c d Axiom 6: a b - a - b If u , then - u , since c d - c - d a b - a - b a - a b - b 0 0 0 u ( -u ) c d - c - d c - c d - d 0 0 推廣：The set of m n matrices, Mmn, is a vector space.
Ch04_5
Vector Spaces of Matrices
p q Let M 22 { | p, q, r , s R }. Prove that M22 is a vector space. r s
Proof
a b e f Let u and v g h M 22 . c d Axiom 1: a b e f a e b f uv c d g h c g d h u + v is a 2 2 matrix. Thus M22 is closed under addition.
Linear Algebra
Chapter 4 General Vector Spaces
大葉大學 資訊工程系 黃鈴玲
4.1 General Vector Spaces and Subspaces
Our aim in this section will be to focus on the algebraic properties of Rn. We draw up a set of axioms (公理) based on the properties of Rn. Any set that satisfies these axioms will have similar algebraic properties to the vector space Rn.
Note. U只要有加法與純量乘法的封閉性，其他axiom都會滿足。
(跳過)
Ch04_10
Theorem 4.1
Let V be a vector space, v a vector in V, 0 the zero vector of V, c a scalar, and 0 the zero scalar. Then (a) 0v = 0 (b) c0 = 0 (c) (-1)v = -v (d) If cv = 0, then either c = 0 or v = 0. Proof (a) 0v + 0v = (0 + 0)v = 0v (0v + 0v) + (-0v) = 0v + (-0v) 0v + [0v + (-0v)] = 0, 0v + 0 = 0, 0v = 0 (c) (-1)v + v = (-1)v + 1v = [(-1) + 1]v = 0v = 0
p q 隨堂作業：7 W { | p, q, r , s 0}. r s
Ch04_7
Vector Spaces of Functions
Prove that V={ f | f: R R } is a vector space.
(跳過)
Let f, g V, c R. For example: f: R R, f(x)=2x, g: R R, g(x)=x2+1. Axiom 1: f + g is defined by (f + g)(x) = f(x) + g(x). f + g : R R f + g V. Thus V is closed under addition. Axiom 2: cf is defined by (cf)(x) = c f(x). cf : R R cf V. Thus V is closed under scalar multiplication.
(u1 , ..., un ) (v1 , ..., vn ) (u1 v1 , ..., un vn ) c(u1 , ..., un ) (cu1 , ..., cun )
It can be shown that Cn with these two operations is a complex vector space.
Definition
A vector space is a set V of elements called vectors, having operations of addition and scalar multiplication defined on it that satisfy the following conditions. (u, v, and w are arbitrary elements of V, and c and d are scalars.) Closure Axioms (最重要！) 1. The sum u + v exists and is an element of V. (V is closed under addition.) 2. cu is an element of V. (V is closed under scalar multiplication.)
Ch04_3
隨堂作業：14
Definition of Vector Space (continued)
Addition Axioms 3. u + v = v + u (commutative property) 4. u + (v + w) = (u + v) + w (associative property) 5. There exists an element of V, called the zero vector, denoted 0, such that u + 0 = u. 6. For every element u of V there exists an element called the negative of u, denoted -u, such that u + (-u) = 0. Scalar Multiplication Axioms 7. c(u + v) = cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10.1u = u
Definition
Let V be a vector space and U be a nonempty subset of V. If U is a vector space under the operations of addition and scalar multiplication of V it is called a subspace of V.
Ch04_8
Vector Spaces of Functions
(跳過)
Axiom 5: Let 0 be the function such that 0(x) = 0 for every xR. 0 is called the zero function. We get (f + 0)(x) = f(x) + 0(x) = f(x) + 0 = f(x) for every xR. Thus f + 0 = f. 0 is the zero vector. Axiom 6: Let the function –f defined by (-f )(x) = -f (x).
[ f (- f )]( x) f ( x) (- f )( x) f ( x) - [ f ( x)] 0 0( x )
Thus [f + (-f )] = 0, -f is the negative of f. 隨堂作業：13 V={ f | f(x)=ax2+bx+c for some a,b,c R}
Ch04_9
The Complex Vector Space Cn
Let (u1 , ..., un ) be a sequenceof n complex numbers. The set of all such sequencesis denotedCn . Let operationsof addition and scalar multiplica tion (by a complex scalar c) be defined on Cn as follows :