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Vector Space Properties
| Content Provider | Scilit |
|---|---|
| Author | Effinger, Gove Mullen, Gary L. |
| Copyright Year | 2019 |
| Description | Book Name: An Elementary Transition to Abstract Mathematics |
| Abstract | Having learned the definition of a vector space and seen numerous examples, let’s now take a close look at some of their most important properties. We start with a simple example. Example 31.1 In Chapter 30 we started with the two vectors v and w in R 2 which are pictured there. Two key questions we can ask about them and which we shall explore and be able to generalize in this chapter are: Do v and w “depend” on each other in some way, or are they “independent?” Given any vector u in R 2 , can we write u in terms of v and w? Dealing first with Question 1, by “depend” we mean: Can one of them, say w, be written in the form λ v for some scalar λ in R ? The answer here is obviously no, since, geometrically, λ v lies on the line through the origin on which v lies, and w is not on that line. Hence we say u and w are “independent.” Now with Question 2, by the words “in terms of” we mean: Can an arbitrary vector u = (s, t) in R 2 be written in the form u = λ v + μ w for some scalars λ and μ in R ? This is called writing u as a linear combination of v and w. Working algebraically now, since v = (2, 1.5) and w = ( − 1, 4), we seek λ and μ by solving two simultaneous linear equations (one involving the first coordinates of v and w; the second involving their second coordinates) as follows: s = 2 λ + ( − 1 ) μ and t = 1.5 λ + 4 μ . Work through this solution (Do it! Good practice). You should arrive at λ = 2 7 s + 1 14 t and μ = − 3 7 s + 1 7 t . 212For these values of λ and μ, we have u = λ v + μ w , and since u was arbitrary, we have established the answer to Question 2. We say then that the independent vectors v and w span all of R 2 . Staying in the vector space R 2 over R but generalizing, let’s assume that u = (s, t), v = (a, b) and w = (c, d) are all arbitrary vectors in R 2 and ask under what conditions is u a linear combination of v and w? This means we seek scalars λ and μ such that s = λ a + μ b and t = λ c + μ b . Solving these simultaneous equations (very carefully), and labeling the quantity ad − bc as D, we arrive at λ = d D s − b D t and μ = − c D s + a D t . We observe then that there are solutions for λ and μ if and only if D = ad − bc is non-zero! Does the quantity ad − bc look familiar? It should, since, as we learned in Chapter 12, it is the determinant of the matrix ( a b c d ) ; that is, the matrix whose rows are the coordinates of the vectors v and w. Hence we get solutions for λ and μ if and only if that matrix is non-singular (i.e., if its determinant is non-zero). In this case the rows of the matrix (and so the vectors they represent) are said to be linearly independent. All this carries over to general vector spaces, as we shall now see. Let v 1 , … , v n be a set of n vectors in a vector space V defined over a field F. We say that this set of vectors is linearly dependent if there are scalars λ 1 , … , λ n ∈ F , not all zero, so that λ 1 v 1 + ⋯ + λ n v n = 0 . If the set is not linearly dependent, it is linearly independent . Equivalently, the set of vectors is linearly independent over the field F if, whenever λ 1 v 1 + ⋯ + λ n v n = 0 , it must be the case that λ$ _{1}$ = · · · = λ$ _{ n }$ = 0. |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2019-0-00596-8&isbn=9780429324819&doi=10.1201/9780429324819-31&format=pdf |
| Ending Page | 215 |
| Page Count | 5 |
| Starting Page | 211 |
| DOI | 10.1201/9780429324819-31 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2019-11-05 |
| Access Restriction | Open |
| Subject Keyword | Book Name: An Elementary Transition to Abstract Mathematics History and Philosophy of Science |
| Content Type | Text |
| Resource Type | Chapter |
