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la_m8_handouts - MAC 2103 Module 8 General Vector Spaces I...

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1 1 MAC 2103 Module 8 General Vector Spaces I 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Recognize from the standard examples of vector spaces, that a vector space is closed under vector addition and scalar multiplication. 2. Determine if a subset W of a vector space V is a subspace of V. 3. Find the linear combination of a finite set of vectors. 4. Find W = span(S), a subspace of V, given a set of vectors S in a vector space V. 5. Determine if a finite set of non-zero vectors in V is a linearly dependent set or linearly independent set. 6. Use the Wronskian to determine if a set of vectors that are differentiable functions is linearly independent. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.
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2 3 Rev.09 General Vector Spaces I http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Real Vector Spaces or Linear Spaces Real Vector Spaces or Linear Spaces Subspaces Subspaces Linear Independence Linear Independence There are three major topics in this module: 4 Rev.F09 What are the Standard Examples of Vector Spaces? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. We have seen some of them before; some standard examples of vector spaces are as follows: Can you identify them? We will look at some of them later in this module. For now, know that we can always add any two vectors and multiply all vectors by a scalar within any vector space . R 1 , R 2 , R 3 , R n , M m , n , P n , C ( !" , " ), C [ a , b ]
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3 5 Rev.F09 What are the Standard Examples of Vector Spaces? (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Since we can always add any two vectors and multiply all vectors by a scalar in any vector space, we say that a vector space is closed under vector addition and scalar multiplication. In other words, it is closed under linear combinations. A vector space is also called a linear space. In fact, a linear space is a better name. 6 Rev.F09 What is a Vector Space? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Let V be a non-empty set of objects u , v , and w , on which two operations, vector addition and scalar multiplication, are defined. If V can satisfy the following ten axioms, then V is a vector space. (Please pay extra attention to axioms 1 and 6.) 1. If u , v V, then u + v V ~ Closure under addition 2. u + v = v + u ~ Commutative property 3. u + ( v + w )= ( u + v )+ w ~ Associative property 4. There is a unique zero vector such that u + 0 = 0 + u = u , for all u in V. ~ Additive identity 5. For each u , there is a unique vector - u such that u + ( -u ) = 0 . ~ Additive inverse
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4 7 Rev.F09 What is a Vector Space? (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Here are the next five properties:
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This note was uploaded on 10/18/2011 for the course MAS 2103 taught by Professor Shaw during the Summer '11 term at Valencia.

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la_m8_handouts - MAC 2103 Module 8 General Vector Spaces I...

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