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College Algebra Exam Review 363

College Algebra Exam Review 363 - 2 – by– 2 submatrix...

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8.4. FINITELY GENERATED MODULES OVER A PID, PART I 373 For example, for m D 4 , D.3; / D 2 6 6 4 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 3 7 7 5 : The third type of elementary row operation interchanges two rows. The operation of interchanging the i –th and j –th rows is implemented by multiplication on the left by the m –by- m permutation matrix P i;j corre- sponding to the transposition .i; j / . P i;j is its own inverse in Mat m .R/ . For example, for m D 4 , P 2;4 D 2 6 6 4 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 3 7 7 5 : When we work over an arbitrary PID R , we require one more type of row operation. In this fourth type of row operation, each of two rows is simultaneously replaced by linear combinations of the two rows. Thus a i is replaced by ˛a i C ˇa j , while a j is replaced by a i C ıa j . We require that this operation be invertible, which is the case precisely when the matrix ˛ ˇ ı is invertible in Mat 2 .R/ . Consider the m –by– m matrix U. ˛ ˇ ı I i; j / that coincides with the identity matrix except for the 2 by– 2 submatrix in the
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Unformatted text preview: 2 – by– 2 submatrix in the i –th and j –th rows and i –th and j –th columns, which is equal to ± ˛ ˇ ± ı ² . For example, when m D 4 , U. ± ˛ ˇ ± ı ² I 2;4/ D 2 6 6 4 1 0 0 0 0 ˛ 0 ˇ 0 0 1 0 0 ± 0 ı 3 7 7 5 : The matrix U. ± ˛ ˇ ± ı ² I i;j/ is invertible with inverse U. ± ˛ ˇ ± ı ² ± 1 I i;j/ . Left multiplication by U. ± ˛ ˇ ± ı ² I i;j/ implements the fourth type of ele-mentary row operation. Elementary column operations are analogous to elementary row oper-ations. They are implemented by right multiplication by invertible n –by– n matrices. We say that two matrices are row–equivalent if one is transformed into the other by a sequence of elementary row operations; likewise, two matrices are column–equivalent if one is transformed into the other by a...
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