Module 1: Basics and Background Lecture 5: Vector and Metric Spaces. The Lecture Contains: Definition of vector space.
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1 The Lecture Contains: Definition of vector space Dimensionality Definition of metric space file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_1.htm[6/14/2012 3:18:58 PM]
2 Definition of vector space Informally, a collection of vectors that can be added together and scaled by a scalar Vector space over field defines two operations Vector addition:, denoted as Scalar multiplication:, denoted as If scalars are real numbers, then is called a real vector space file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_2.htm[6/14/2012 3:18:59 PM]
3 Definition of vector space Informally, a collection of vectors that can be added together and scaled by a scalar Vector space over field defines two operations Vector addition:, denoted as Scalar multiplication:, denoted as If scalars are real numbers, then is called a real vector space 8 properties of addition and : Commutativity of vector addition: Associativity of vector addition: Additive identity: (zero vector), s.t. Additive inverse:, s.t. Multiplicative identity: (multiplicative identity of F), s.t. Associativity of scalar multiplication: Distributivity of scalar sums: Distributivity of vector sums: file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_3.htm[6/14/2012 3:18:59 PM]
4 file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_4.htm[6/14/2012 3:18:59 PM]
5 Are the following subspaces? Zero vector: file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_5.htm[6/14/2012 3:18:59 PM]
6 Are the following subspaces? Zero vector: Yes file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_6.htm[6/14/2012 3:18:59 PM]
7 Are the following subspaces? Zero vector: Yes First quadrant: file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_7.htm[6/14/2012 3:19:00 PM]
8 Are the following subspaces? Zero vector: Yes First quadrant: No file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_8.htm[6/14/2012 3:19:00 PM]
9 Are the following subspaces? Zero vector: Yes First quadrant: No First and third quadrant: file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_9.htm[6/14/2012 3:19:00 PM]
10 Are the following subspaces? Zero vector: Yes First quadrant: No First and third quadrant: No file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_10.htm[6/14/2012 3:19:00 PM]
11 Are the following subspaces? Zero vector: Yes First quadrant: No First and third quadrant: No Lower triangular : file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_11.htm[6/14/2012 3:19:00 PM]
12 Are the following subspaces? Zero vector: Yes First quadrant: No First and third quadrant: No Lower triangular : Yes file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_12.htm[6/14/2012 3:19:00 PM]
13 Are the following subspaces? Zero vector: Yes First quadrant: No First and third quadrant: No Lower triangular : Yes Symmetric : file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_13.htm[6/14/2012 3:19:01 PM]
14 Are the following subspaces? Zero vector: Yes First quadrant: No First and third quadrant: No Lower triangular : Yes Symmetric : Yes file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_14.htm[6/14/2012 3:19:01 PM]
15 Dimensionality Assume there are k vectors What is their dimensionality? file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_15.htm[6/14/2012 3:19:01 PM]
16 Dimensionality Assume there are k vectors What is their dimensionality? Vectors are linearly independent iff the linear combination only when file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_16.htm[6/14/2012 3:19:01 PM]
17 Dimensionality Assume there are k vectors What is their dimensionality? Vectors are linearly independent iff the linear combination only when The span of a set of vectors is the vector space generated by their linear combinations file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_17.htm[6/14/2012 3:19:01 PM]
18 Dimensionality Assume there are k vectors What is their dimensionality? Vectors are linearly independent iff the linear combination only when The span of a set of vectors is the vector space generated by their linear combinations A basis of a vector space is a set of vectors that are linearly independent and that spans file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_18.htm[6/14/2012 3:19:02 PM]
19 Dimensionality Assume there are k vectors What is their dimensionality? Vectors are linearly independent iff the linear combination only when The span of a set of vectors is the vector space generated by their linear combinations A basis of a vector space is a set of vectors that are linearly independent and that spans The cardinality of the basis of a vector space, i.e., the number of linearly independent vectors needed to span is called its dimensionality The bases (i.e., the basis vectors) may vary, but their cardinality remains the same file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_19.htm[6/14/2012 3:19:02 PM]
20 Definition of metric space A set of elements with a distance function defined between any two elements of the set 4 properties of d: Non-negativity: Identity: Symmetry: Triangular inequality: file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_20.htm[6/14/2012 3:19:02 PM]
21 Definition of metric space A set of elements with a distance function defined between any two elements of the set 4 properties of d: Non-negativity: Identity: Symmetry: Triangular inequality: Pseudometric space: Condition 2 (identity) is relaxed Example: Number of vertices and edges of a graph Quasimetric space: Condition 3 (symmetry) is relaxed Example: Time to walk from plain A to hill B Semimetric space: Condition 4 (triangular inequality) is relaxed Example: Air fares on certain routes file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture5/5_21.htm[6/14/2012 3:19:02 PM]
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