Abstract Algebra Theory and Applications - Computer Science ...

172 CHAPTER 10 MATRIX GROUPS AND SYMMETRYthenT (x) = T (x 1 e 1 + x 2 e 2 + · · · + x n e n )= x 1 T (e 1 ) + x 2 T (e 2 ) + · · · + x n T (e n )( n∑) tn∑= a 1k x k , . . . , a mk x kk=1= Ax.k=1Example 1. If we let T : R 2 → R 2 be the map given byT (x 1 , x 2 ) = (2x 1 + 5x 2 , −4x 1 + 3x 2 ),the axioms that T must satisfy to be a linear transformation are easilyverified. The column vectors T e 1 = (2, −4) t and T e 2 = (5, 3) t tell us thatT is given by the matrixA =( 2 5−4 3Since we are interested in groups of matrices, we need to know whichmatrices have multiplicative inverses. Recall that an n × n matrix A isinvertible exactly when there exists another matrix A −1 such that AA −1 =A −1 A = I, where⎛⎞1 0 · · · 00 1 · · · 0I = ⎜⎝.. . ..⎟. ⎠0 0 · · · 1is the n×n identity matrix. From linear algebra we know that A is invertibleif and only if the determinant of A is nonzero. Sometimes an invertiblematrix is said to be nonsingular.Example 2. If A is the matrix( 2 15 3),).then the inverse of A isA −1 =( 3 −1−5 2).