Linear Algebra

Section IV. Matrix Operations 211Distribute and regroup on the v’s.= (g i,1 h 1,1 + g i,2 h 2,1 + · · · + g i,r h r,1 ) · v 1+ · · · + (g i,1 h 1,n + g i,2 h 2,n + · · · + g i,r h r,n ) · v nFinish by recognizing that the coefficient of each v jg i,1 h 1,j + g i,2 h 2,j + · · · + g i,r h r,jmatches the definition of the i, j entry of the product GH.QEDThe theorem is an example of a result that supports a definition. We canpicture what the definition and theorem together say with this arrow diagram(‘wrt’ abbreviates ‘with respect to’).W wrt ChgHGg ◦ hV wrt B X wrt DGHAbove the arrows, the maps show that the two ways of going from V to X,straight over via the composition or else by way of W , have the same effect⃗v ↦−→ g◦hg(h(⃗v)) ⃗v ↦−→ hgh(⃗v) ↦−→ g(h(⃗v))(this is just the definition of composition). Below the arrows, the matrices indicatethat the product does the same thing — multiplying GH into the columnvector Rep B (⃗v) has the same effect as multiplying the column first by H andthen multiplying the result by G.Rep B,D (g ◦ h) = GH = Rep C,D (g) Rep B,C (h)The definition of the matrix-matrix product operation does not restrict usto view it as a representation of a linear map composition. We can get insightinto this operation by studying it as a mechanical procedure. The striking thingis the way that rows and columns combine.One aspect of that combination is that the sizes of the matrices involved issignificant. Briefly, m×r times r×n equals m×n.2.7 Example This product is not defined( ) ( )−1 2 0 0 00 10 1.1 0 2because the number of columns on the left does not equal the number of rowson the right.