lecture_28 - MA 265 LECTURE NOTES FRIDAY MARCH 21 Row and...

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MA 265 LECTURE NOTES: FRIDAY, MARCH 21 Row and Column Space Rank of a Matrix. Let A be an m × n matrix. We have seen that the row rank of A is the dimension of the subspace of R n spanned by the rows of A ; and that the column space of A is the dimension of the subspace of R m spanned by the columns of A . We show that the row rank is equal to the column rank. To be explicit, let’s write A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a mn . and denote the i th row and the j th column of A as the matrices row i ( A ) = ± a i 1 a i 2 ··· a in ² R n and col j ( A ) = a 1 j a 2 j . . . a mj R m . Let r be the row rank of A , and say that S = { w 1 , w 2 , ..., w r } is a basis for the row space of A . Then each row can be expressed as a linear combination of these basis vectors: row 1 ( A ) = c 11 w 1 + c 12 w 2 + ··· + c 1 r w r row 2 ( A ) = c 21 w 1 + c 22 w 2 + ··· + c 2 r w r . . . row m ( A ) = c m 1 w 1 + c m 2 w 2 + ··· + c mr w r Upon writing w i = ± b i 1 b i 2 ··· b in ² and looking at the j th columns, we have the following expressions: a ij = c 11 b 1 j + c 12 b 2 j + ··· + c 1 r b rj a 2 j = c 21 b 1 j + c 22 b 2 j + ··· + c 2 r b rj . . . a
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lecture_28 - MA 265 LECTURE NOTES FRIDAY MARCH 21 Row and...

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