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P1_130dd

# P1_130dd - Review sheet for Math 415.01 Midterm I A matrix...

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Review sheet for Math 415.01 Midterm I A matrix consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. The element in the i th row and j th column is designated a ij and the matrix can be written as ( a ij ). We associate with each matrix A several other matrices: the transpose of the matrix A T = ( a ij ) T = ( a ji ), wherein we interchange rows and columns, the complex conjugate, A , wherein we replace each element by its complex conjugate, and the adjoint, A * = A T , wherein we do both of the above. One adds two matrices or multiples a matrix by a scalar (i.e. a complex number) in the obvious term-by-term manner. If A = ( a ij ) is an m × n matrix and B = ( b k‘ ) is an n × r matrix, then AB = ( n s =1 a is b sj ). That is to say, each element of the product matrix is the dot product of a row of the first factor and a column of the second The dot product of two vectors, usually written ( x , y ) or < x , y > , can be written in terms of matrix multiplication as x T y (we consider vectors as matrices with one column).

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