Here we consider the matrix transformation for a square matrix from a geometric point of view.

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1 Section. he Mgnifiction Fctor In Section.5 we iscusse the mtri trnsformtion etermine mtri A. For n m n mtri A the function f(c) = Ac provies corresponence etween vectors in R n n R m. Here we consier the mtri trnsformtion for squre mtri from geometric point of view. Initill we tke A to e n etermine how the mtri trnsformtion f(c) = Ac chnges the re of close polgon. In Emple 8 of Section.5 we showe tht the imge of the unit squre when trnsforme the igonl mtri h 0 ws rectngle with sies of length h n k. 0 k In Eercise 4 of Section.5 ou were ske to show tht the imge of the unit squre trnsforme the mtri is prllelogrm n c to show how to compute the re of the prllelogrm. In Emple we etermine the imge of n ritrr rectngle n compute its re. Emple. Let A = c. o fin the imge of the rectngle R shown in Figure the mtri trnsformtion etermine A we follow the proceure use in Emple 8 of Section.5. Figure.

2 Let S e the mtri whose columns re the vertices of the rectngle S = f(s) = AS =. hen the imge of the mtri trnsformtion is c = + 0 c+ c 0 he vertices in f(s) efine the prllelogrm which is shown in Figure.. Figure. Net we etermine the reltionship etween the res of the rectngle R in Figure n the prllelogrm P in Figure. From Figure we see tht i) he re of the two rectngles lele, is (c)(). ii) he re of the two tringles lele,, is ()(). iii) he re of the two tringles lele,,, is ()(c). iv) he re of the rectngle enclosing P n the regions lele,,,,, n,,, is ( + )(c + ). Figure.

3 hus re(p) = ( + )(c + ) - (c)() - ()() - ()(c) = - c so the re of the imge P is - c re(r). It follows tht re(p) c = = c. () re(r) From () we see tht the fctor which the re of the imge chnges epens onl on the entries of the mtri A. We cll this the mgnifiction fctor of the mtri trnsformtion etermine A. A nturl question is, Does the mgnifiction fctor for mtri trnsformtion f(c) = Ac chnge if we chnge the geometric figure? o emine this question we investigte severl cses for fmilir figures.

4 Emple. Let e the tringle with vertices P, Q, n R in Figure 4. he re of tringle is liner comintion of the res of trpezois APQB, BQRC, n APRC s follows: re() = re(apqb) + re(bqrc) - re(aprc). he re of trpezoi is times 'its height' times 'the sum of the lengths of its prllel sies'. re(apqb) = re(bqrc) = re(aprc) = (+ ) Sustituting these epressions into () n collecting terms we hve (verif) re() = + + = Figure o mke this epression inepenent of the orer in which the vertices of the tringle re lele we tke the solute vlue of the right sie of the preceing epression: re() =. ()

5 Before we compute the mgnifiction fctor for the mtri trnsformtion etermine A = in n equivlent vector form. Let C = hen c, it is convenient to epress () = C (see Eercise 5). With = cn e epresse in vector-mtri form s re() C C. n =, () = =. () he imge of the mtri trnsformtion etermine A = c is f( S) = AS= c = c+ c + c +. Geometricll this imge is tringle A with vertices {( +,c + ), ( +,c + ), ( +,c + )} which correspon to the columns of f(s). he re of the imge A is compute using (4) with replce replce c c c c = + 6. hus re( A ) = + C( c + ) = + n = c( C) + C + c C + C.

6 From Eercise 5 we hve C C C C Hence it follows tht A = 0, =, n = 0. ( ) ( ) ( ) = = re( ) = C c C (-c) C (-c) re() herefore the mgnifiction fctor in this cse is lso - c. Emple. Let P e the prllelogrm shown in Figure 5. Since igonl of prllelogrm ivies it into two congruent tringles we hve from (4) tht re(p) = C = C. he imge of P the mtri trnsformtion etermine A = is nother prllelogrm, hence pir of congruent tringles. From Emple it follows tht the re of the imge of P is - c times the re of P. Hence the mgnifiction fctor is gin - c. Net we investigte the mgnifiction fctor for the imge of n ritrr close polgon the mtri trnsformtion etermine A = c Figure 5.. Since n close polgon cn e suivie into set of nonoverlpping tringles, s shown in Figures 6 n 6, we cn use the results estlishe so fr. he process epicte in Figure 6 is c

7 clle tringultion. (Mn ifferent tringultions re possile for given polgon. ) Once we hve performe tringultion we ppl the results of Emple. It follows tht the mgnifiction fctor is gin - c. Emple 4. o etermine the re of the polgon shown in Figure 7 we perform tringultion s shown in Figure 8. Net we use the results of Emple. Figure 7. Figure 8. 'ringultion is the prtitioning of polgon into nonoverlpping tringles ll of whose vertices re vertices of the polgon.', he Wors of Mthemtics S. Schwrtzmn, he MAA, Wshington, D.C., 994. Note tht n ege is shre the joining tringles.

8 re( ) = hus we hve 0 C 0.5 re( ) = = re( ) = C.5 = 0 C 4.5 = re(polgon in Figure 7) = re( ) + re( ) + re( ) = 9.5 sq. units. A remining question is, Wht is the mgnifiction fctor if the geometric figure is nonpolgonl region? ht is, close region with curve ounries. Mn nonpolgonl region cn e pproimte s closel s esire polgon, possil with mn sies. See Emple 5. hus it follows tht the mgnifiction fctor for the mtri trnsformtion etermine A = c of nonpolgonl region will lso e - c. Emple 5. A circle cn e pproimte s closel s esire inscrie polgons s inicte in Figures 9,, n c Figure 9. Figure 9. Figure 9c.

9 In Emple in Section.4 we showe tht mtri A = ws nonsingulr provie - c 0 n then A = c fctor for the mtri trnsformtion etermine A = c is c c. From our evelopment in this section we hve tht the mgnifiction - c. Geometricll, we cn see tht the mtri trnsformtion is 'reversile' if n onl if A is nonsingulr. he mgnifiction fctor of mtri trnsformtion etermine squre mtri A is n intrinsic numer ssocite with A.