Linear Algebra

Section III. Computing Linear Maps 201for each i.(d) Conclude that for every linear map h: V → W there are bases B, D so thematrix representing h with respect to B, D is upper-triangular (that is, eachentry h i,j with i > j is zero).(e) Is an upper-triangular representation unique?III.2Any Matrix Represents a Linear MapThe prior subsection shows that the action of a linear map h is described by amatrix H, with respect to appropriate bases, in this way.⎛ ⎞v 1⎜ ⎟⃗v = ⎝ . ⎠v nB⎛⎞h 1,1 v 1 + · · · + h 1,n v nh⎜⎟↦−→ h(⃗v) = ⎝H. ⎠h m,1 v 1 + · · · + h m,n v nHere we will show the converse, that each matrix represents a linear map.So we start with a matrix⎛⎞h 1,1 h 1,2 . . . h 1,nh 2,1 h 2,2 . . . h 2,nH =⎜⎟⎝ .⎠h m,1 h m,2 . . . h m,nand we will describe how it defines a map h. We require that the map berepresented by the matrix so first note that in (∗) the dimension of the map’sdomain is the number of columns n of the matrix and the dimension of thecodomain is the number of rows m. Thus, for h’s domain fix an n-dimensionalvector space V and for the codomain fix an m-dimensional space W. Also fixbases B = 〈⃗β 1 , . . . , ⃗β n 〉 and D = 〈⃗δ 1 , . . . ,⃗δ m 〉 for those spaces.Now let h: V → W be: where ⃗v in the domain has the representation⎛ ⎞v 1⎜ ⎟Rep B (⃗v) = ⎝ . ⎠v nthen its image h(⃗v) is the member the codomain with this representation.⎛⎞h 1,1 v 1 + · · · + h 1,n v n⎜⎟Rep D ( h(⃗v) ) = ⎝ . ⎠h m,1 v 1 + · · · + h m,n v nThat is, to compute the action of h on any ⃗v ∈ V, first express ⃗v with respect tothe basis ⃗v = v 1⃗β 1 + · · · + v n⃗β n and then h(⃗v) = (h 1,1 v 1 + · · · + h 1,n v n ) · ⃗δ 1 +· · · + (h m,1 v 1 + · · · + h m,n v n ) · ⃗δ m .BDD(∗)