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lecture_5 - MA 35100 LECTURE NOTES FRIDAY JANUARY 22 Rank...

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MA 35100 LECTURE NOTES: FRIDAY, JANUARY 22 Rank of a Matrix Say we have a system of equations. Denote A as its augmented matrix. We may compute the reduced row-echelon form E = rref( A ), and use it to solve this system by considering the pivots. We solve for the leading variables in terms of the free variables. Of course, we can rearrange the equations so that sometimes we can express the free variables in terms of the leading variables. Ultimately what matters is the number of leading variables, not really which leading variables we have. We use this to make a definition. Definition. The rank of a matrix A is the number of pivots in rref( A ). We denote this nonnegative integer by rank( A ). We give a brief example. Consider the following matrix: A = 1 2 3 4 5 6 7 8 9 . We compute that the reduced row-echelon form for this matrix is E = rref( A ) = 1 0 - 1 0 1 2 0 0 0 . Note that this matrix has two pivots (which are circled above), so that the rank is rank( A ) = 2. We note that we have defined the rank of a matrix , not the rank of a system of equations. This definition is a but more subtle, so we defer more discussion until a later lecture.

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