Linear Algebra

Section III. Computing Linear Maps 191IIIComputing Linear MapsThe prior section shows that a linear map is determined by its action on a basis.The equationh(⃗v) = h(c 1 · ⃗β 1 + · · · + c n · ⃗β n ) = c 1 · h(⃗β 1 ) + · · · + c n · h(⃗β n )describes how we get the value of the map on any vector ⃗v by starting from thevalue of the map on the vectors ⃗β i in a basis and extending linearly.This section gives a convenient scheme to use the representations of h(⃗β 1 ),. . . , h(⃗β n ) to compute, from the representation of a vector in the domainRep B (⃗v), the representation of that vector’s image in the codomain Rep D (h(⃗v)).III.1Representing Linear Maps with Matrices1.1 Example For the spaces R 2 and R 3 fix⎛( ) ( )2 1⎜1⎞ ⎛ ⎞ ⎛0⎟ ⎜ ⎟ ⎜1⎞⎟B = 〈 , 〉 and D = 〈 ⎝0⎠ , ⎝−2⎠ , ⎝0⎠〉0 40 0 1as the bases. Consider the map h: R 2 → R 3 with this action.⎛ ⎞( ) 12 h ⎜ ⎟↦−→ ⎝1⎠01⎛( )1 h ⎜1⎞⎟↦−→ ⎝2⎠40To compute the action of this map on any vector at all from the domain we firstexpress, with respect to the codomain’s basis, h(⃗β 1 )⎛⎜1⎞ ⎛⎟ ⎜1⎞ ⎛ ⎞ ⎛⎟⎝1⎠ = 0 ⎝0⎠ − 1 0⎜ ⎟ ⎜1⎞⎛ ⎞0⎟⎜ ⎟⎝−2⎠ + 1 ⎝0⎠ so Rep2D (h(⃗β 1 )) = ⎝−1/2⎠1 0 0 11Dand h(⃗β 2 ).⎛⎜1⎞ ⎛⎟ ⎜1⎞ ⎛ ⎞ ⎛0⎟ ⎜ ⎟ ⎜1⎞⎛ ⎞1⎟⎜ ⎟⎝2⎠ = 1 ⎝0⎠ − 1 ⎝−2⎠ + 0 ⎝0⎠ so Rep D (h(⃗β 2 )) = ⎝−1⎠0 0 0 10D