hw2 - Math 205 Spring 2006 B Dodson Week 2 Ax = 0 homog...

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Math 205, Spring 2006 B. Dodson Week 2: Ax = 0 , homog.; Matrix Inverse. The Rank of a matrix is defined to be the number of non-zero rows in the RREF of the matrix. We do not necessarily need the RREF or even a REF to find the rank; for example, a (square) upper-triagular matrix that is n × n with non-zero diagonal entries always has rank n. (why?) We will discuss the results of the qualitative theory more later on the text; but the first main result is that when Ax = b, with A an m × n matrix, has rank( A ) = rank( A # ) = n, the system has a unique solution. (We’re using A # = ( A | b ) for the augmented matrix.)
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2 Next, whenever rank( A ) = rank( A # ) we may read-off a particular solution x p , so that the system is consistent; while rank( A ) = rank( A # ) occurs only when rank( A # ) = rank( A ) +1, in which case the last equation reads 0 = 1 , which is inconsistent, so the system has no solution. Finally, if r = rank( A ) = rank( A # ) < n, there are infinitely many solutions; and with r leading 1’s; so r bound variables, there are d = n - r free varibles and d linearly independent solutions to the homog. equation.
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