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In Application 2, he's talking about searching databases. He says we should imagine a database of m documents and n possible search words. Then he says that this can be put into an mxn matrix in which each column represents a book-- the jth entry of the column is 1 if that book contains the word, 0 if it doesn't.

He then says a "search vector" lives in R^m, and it's a column vector whose jth entry is 1 if you are seraching for that word.

Does he have his m's and n's mixed up? It seems to me like if you have m documents and each column of the matrix represents one, you need an nxm matrix. Similarly, you can't search through n words by using a vector in R^m, correct? Does he mean you take an nxm matrix and multiply its transpose by the vector in R^n?