Apprentice Linear Algebra , 4 th lecture , 07 / 08 / 05 REU 2005 Instructor : László Babai Scribe :
نویسنده
چکیده
a system has a solution. Recall that if x = x1 .. x` , and the matrix A is written as A = [ a1 . . . a` ] , where ai is the ith column of the matrix, then Ax = ∑i=` i=1 xiai is simply a linear combination of the columns. Therefore if Ax = b holds, then b ∈ Span{a1, . . . ,a`}. Recall that the span of the ai is called the column space of the matrix A. It follows that rk(A) = rk ([A|b]), where the augmented matrix [A|b] is defined as [ a1 . . . a` b ]. Conversely, if we know rk(A) = rk ([A|b]), then b is in the column space of A. We have thus shown the following:
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