Linear Algebra

220 Chapter Three. Maps Between Spaces(This is the representation with respect to E 2 , E 2 , the standard basis, of a rotationthrough π/6 radians counterclockwise.)2.36 The infinite-dimensional space P of all finite-degree polynomials gives a memorableexample of the non-commutativity of linear maps. Let d/dx: P → P be theusual derivative and let s: P → P be the shift map.a 0 + a 1 x + · · · + a n x n s↦−→ 0 + a 0 x + a 1 x 2 + · · · + a n x n+1Show that the two maps don’t commute d/dx ◦ s ≠ s ◦ d/dx; in fact, not only is(d/dx ◦ s) − (s ◦ d/dx) not the zero map, it is the identity map.2.37 Recall the notation for the sum of the sequence of numbers a 1 , a 2 , . . . , a n .n∑a i = a 1 + a 2 + · · · + a ni=1In this notation, the i, j entry of the product of G and H is this.r∑p i,j = g i,k h k,jk=1Using this notation,(a) reprove that matrix multiplication is associative;(b) reprove Theorem 2.6.IV.3Mechanics of Matrix MultiplicationIn this subsection we consider matrix multiplication as a mechanical process,putting aside for the moment any implications about the underlying maps.The striking thing about matrix multiplication is the way rows and columnscombine. The i, j entry of the matrix product is the dot product of row i of theleft matrix with column j of the right one. For instance, here a second row anda third column combine to make a 2, 3 entry.⎛ ⎞1 1 (⎜ ⎟ 4⎝ 0 1 ⎠51 06789)2=3⎛⎞9 13 17 5⎜⎟⎝5 7 9 3⎠4 6 8 2We can view this as the left matrix acting by multiplying its rows, one at a time,into the columns of the right matrix. Or, another perspective is that the rightmatrix uses its columns to act on the left matrix’s rows. Below, we will examineactions from the left and from the right for some simple matrices.The action of a zero matrix is easy.3.1 Example Multiplying by an appropriately-sized zero matrix from the left orfrom the right results in a zero matrix.( ) () ( ) ( ) ( ) ( )0 0 1 3 2 0 0 0 2 3 0 0 0 0==0 0 −1 1 −1 0 0 0 1 4 0 0 0 0The next easiest to understand matrices, after the zero matrices, are the oneswith a single nonzero entry.