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Lect2-02-web

Lect2-02-web - MATH 304 Fall 2011 Linear Algebra Homework...

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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #4 (due Thursday, September 29) All problems are from Leon’s book (8th edition). Section 3.1: 9a, 9b, 11 Section 3.2: 1b, 1c, 11b, 11c, 13a, 13b, 20 MATH 304 Linear Algebra Part II ( ≈ 4.5 weeks): Abstract linear algebra • Vector spaces • Linear independence • Basis and dimension • Coordinates, change of basis • Linear transformations Leon’s book : Chapters 3–4 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 ) mapsto→ x + y ∈ V and scalar multiplication R × V ∋ ( r , x ) mapsto→ r x ∈ 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 , such that a + = + 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 = . A5. r ( a + b ) = r a + r b for all r ∈ R and a , b ∈ V . A6. ( r + s ) a = r a + s a for all r , s ∈ R and a ∈ V . A7. ( rs ) a = r ( s a ) for all r , s ∈ R and a ∈ V . A8. 1 a = a for all a ∈ V . Examples of vector spaces • R n : n-dimensional coordinate vectors • M m , n ( R ): m × n matrices with real entries • R ∞ : infinite sequences ( x 1 , x 2 , . . . ), x i ∈ R • { } : the trivial vector space • F ( R ): the set of all functions f : R → R • C ( R ): all continuous functions f : R → R • C 1 ( R ): all continuously differentiable functions f : R → R • C ∞ ( R ): all smooth functions f : R → R • P : all polynomials p ( x ) = a + a 1 x + ··· + a n x n Subspaces of vector spaces Definition. A vector space V is a subspace of a vector space V if V ⊂ V and the linear operations on V agree with the linear operations on V . Proposition A subset S of a vector space V is a subspace of V if and only if S is nonempty and closed under linear operations , i.e., x , y ∈ S = ⇒ x + y ∈ S , x ∈ S = ⇒ r x ∈ S for all r ∈ R ....
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Lect2-02-web - MATH 304 Fall 2011 Linear Algebra Homework...

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