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For two sets of strings S 1 and S 2, the concatenation S 1 S 2 consists of all strings of the form vw where v is a string from S 1 and w is a string from S 2, or formally S 1 S 2 = { vw : v ∈ S 1, w ∈ S 2}. Many authors also use concatenation of a string set and a single string, and vice versa, which are defined similarly by S 1 w = { vw ...
For example, a two-dimensional array A with three rows and four columns might provide access to the element at the 2nd row and 4th column by the expression A[1][3] in the case of a zero-based indexing system. Thus two indices are used for a two-dimensional array, three for a three-dimensional array, and n for an n-dimensional array.
The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
For example, for the 2×2 matrix = [], the half-vectorization is = []. There exist unique matrices transforming the half-vectorization of a matrix to its vectorization and vice versa called, respectively, the duplication matrix and the elimination matrix .
An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts.For example, a 2,1 represents the element at the second row and first column of the matrix.
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The following is an excerpt from the latest edition of Yahoo's fantasy football newsletter, Get to the Points! If you like what you see, you can subscribe for free here. Most fantasy advice will ...
Popular Republican Gov. Chris Sununu of New Hampshire, who has long been a vocal Trump critic, says there's still room in the GOP for those outside the MAGA and America First base.