Abstract Algebra Theory and Applications - Computer Science ...

10.1 MATRIX GROUPS 171as column matrices, then an m × n matrix⎛A = ⎜⎝⎞a 11 a 12 · · · a 1na 21 a 22 · · · a 2n.. . ..⎟. ⎠a m1 a m2 · · · a mnmaps the vectors to R m linearly by matrix multiplication. Observe that ifα is a real number,A(x + y) = Ax + AyαAx = A(αx),where⎛x = ⎜⎝⎞x 1x 2⎟.x n⎠ .We will often abbreviate the matrix A by writing (a ij ).Conversely, if T : R n → R m is a linear map, we can associate a matrixA with T by considering what T does to the vectorse 1 = (1, 0, . . . , 0) te 2 = (0, 1, . . . , 0) t.e n = (0, 0, . . . , 1) t .We can write any vector x = (x 1 , . . . , x n ) t asConsequently, ifx 1 e 1 + x 2 e 2 + · · · + x n e n .T (e 1 ) = (a 11 , a 21 , . . . , a m1 ) t ,T (e 2 ) = (a 12 , a 22 , . . . , a m2 ) t ,.T (e n ) = (a 1n , a 2n , . . . , a mn ) t ,