In Application 2, suppose that we are searching the database of seven linear algebra books for the search words elementary, matrix, algebra. Form a search vector x, and then compute a vector y that represents the results of the search. Explain the significance of the entries of the vector y.
Application 2:
The growth of digital libraries on the Internet has led to dramatic improvements in thestorage and retrieval of information. Modern retrieval methods are based on matrixtheory and linear algebra.
In a typical situation, a database consists of a collection of documents and we wishto search the collection and find the documents that best match some particular searchconditions. Depending on the type of database, we could search for such items asresearch articles in journals, Web pages on the Internet, books in a library, or moviesin a film collection.
To see how the searches are done, let us assume that our database consists of mdocuments and that there are n dictionary words that can be used as keywords forsearches. Not all words are allowable, since it would not be practical to search forcommon words such as articles or prepositions. If the key dictionary words are orderedalphabetically, then we can represent the database by an m ? n matrix A. Eachdocument is represented by a column of the matrix. The first entry in the j th columnof A would be a number representing the relative frequency of the first key dictionaryword in the j th document. The entry a2 j represents the relative frequency of the secondword in the j th document, and so on. The list of keywords to be used in the searchis represented by a vector x in Rm. The ith entry of x is taken to be 1 if the ith wordin the list of keywords is on our search list; otherwise, we set xi = 0. To carry out thesearch, we simply multiply AT times x.
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