combinepdf-min-0320.pdf

# combinepdf-min-0320.pdf - Notes on linear algebra(Monday...

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Unformatted text preview: Notes on linear algebra (Monday 17th October, 2016, 23:10) page 167 Deﬁnition 3.141. Let n E N. Leta and o be two distinct elements of {1,2, . . . , a}. Consider the transformation which transforms an a X m-matrix C (for some at E IN) into the product Tuer, where Tm is the swapping matrix as deﬁned in Definition 3.102. It is the transformation that modifies an a >< m-matrix C by swapping the a-th row with the v-th row (according to Proposition 3.105). This transformation will be called the row swap Tm. (We are using the same symbol Tum for the transformation and for the swapping matrix.) Note that two matrices C and D (with 71 rows each) satisfy C it} D if and only if they satisfy D = Tuer. (This is because the row swap Turf, transforms each a x m-matrix C into 1],.er .) Note that the row swap Tm is inverse to itself (since the matrices Tao and TM. Ta :3 are inverse). Hence. if two matrices C and D (with a rows each) satisfy C —’> D, Tum then they satisfy C : D. Tam Example 3.142. The row swap T1.3 modifies a 4 X m-matrix C by swapping the 1-st row of C with the 3-rd row of C. Thus, this row swap transforms any 4 x 2- :11 b1 (13 b3 . [12 b2 . :12 b2 . . . . matrix mto . Usmg the arrow notatlon, we can wrlte thls £13 193 111 ill ...
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