MATH 304 Linear Algebra（304年数学线性代数）.pdfVIP

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V × V ∋ (x,y) → x + y ∈ V and scalar multiplication R × V ∋ (r ,x) → rx ∈ V , that have the following properties: Properties of addition and scalar multiplication A1. a + b = b + a for all a, b ∈ V . A2. (a + b) + c = a + (b + c) for all a, b, c ∈ V . A3. There exists an element of V , called the zero vector and denoted 0, such that a + 0 = 0 + a = a for all a ∈ V . A4. For any a ∈ V there exists an element of V , denoted −a, such that a + (−a) = (−a) + a = 0. A5. r (a + b) = ra + r b for all r ∈ R and a, b ∈ V . A6. (r + s )a = ra + sa for all r , s ∈ R and a ∈ V . A7. (rs)a = r (sa) for all r , s ∈ R and a ∈ V . A8. 1a = a for all a ∈ V . Examples of vector spaces n • R : n-dimensional coordinate vectors • Mm,n(R): m ×n matrices with real entries • R∞ : infinite sequences (x ,x , . . . ), x ∈ R 1 2 i • {0}: the trivial vector space • F (R): the set of all functions f : R → R • C(R): all continuous functions f : R → R 1 • C (R): all continuously differentiable functions f : R → R • C∞ (R): all smooth functions f : R → R • P : all polynomials p(x ) = a + a x + + a x n 0 1 n Subspaces of vector spaces Definition.