Linear Algebra

238 Chapter Three. Maps Between SpacesTo finish we will show that any nonsingular matrix M performs a change ofbasis operation from any given starting basis B to some ending basis. Because thematrix is nonsingular it will Gauss-Jordan reduce to the identity. If the matrixis the identity I then the statement is obvious. Otherwise there are elementaryreduction matrices such that R r · · · R 1 · M = I with r 1. Elementary matricesare invertible and their inverses are also elementary so multiplying both sides ofthat equation from the left by R −1 r , then by R −1 r−1 , etc., gives M as a product−1 of elementary matrices M = R 1 · · · R −1 r . (We’ve combined R −1 r I to makeR −1 r ; because r 1 we can always make the I disappear in this way, which weneed to do because it isn’t an elementary matrix.)Thus, we will be done if we show that elementary matrices change a givenbasis to another basis, for then R −1 r changes B to some other basis B r , andR −1 r−1 changes B r to some B r−1 , etc., and the net effect is that M changes Bto B 1 . We will prove this by covering the three types of elementary matricesseparately; here are the three cases.⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞c 1 c 1 c 1 c 1⎛ ⎞ ⎛ ⎞c 1 c 1 ......c ic jc ic iM i (k)c i=kc iP i,j .=.C i,j (k).=.⎜ ⎟ ⎜ ⎟⎝ . ⎠ ⎝ . ⎠c jc ic jkc i + c j⎜c n c n ⎝⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟. ⎠ ⎝ . ⎠ ⎝ . ⎠ ⎝ . ⎠c n c n c nApplying a row-multiplication matrix M i (k) changes a representation withrespect to 〈⃗β 1 , . . . , ⃗β i , . . . , ⃗β n 〉 to one with respect to 〈⃗β 1 , . . . , (1/k)⃗β i , . . . , ⃗β n 〉.c n⃗v = c 1 · ⃗β 1 + · · · + c i · ⃗β i + · · · + c n · ⃗β n↦→ c 1 · ⃗β 1 + · · · + kc i · (1/k)⃗β i + · · · + c n · ⃗β n = ⃗vWe can easily see that the second one is a basis, given that the first is a basisand that k ≠ 0 is a restriction in the definition of a row-multiplication matrix.Similarly, left-multiplication by a row-swap matrix P i,j changes a representationwith respect to the basis 〈⃗β 1 , . . . , ⃗β i , . . . , ⃗β j , . . . , ⃗β n 〉 into one with respect tothis basis 〈⃗β 1 , . . . , ⃗β j , . . . , ⃗β i , . . . , ⃗β n 〉.⃗v = c 1 · ⃗β 1 + · · · + c i · ⃗β i + · · · + c j⃗β j + · · · + c n · ⃗β n↦→ c 1 · ⃗β 1 + · · · + c j · ⃗β j + · · · + c i · ⃗β i + · · · + c n · ⃗β n = ⃗vAnd, a representation with respect to 〈⃗β 1 , . . . , ⃗β i , . . . , ⃗β j , . . . , ⃗β n 〉 changes vialeft-multiplication by a row-combination matrix C i,j (k) into a representationwith respect to 〈⃗β 1 , . . . , ⃗β i − k⃗β j , . . . , ⃗β j , . . . , ⃗β n 〉⃗v = c 1 · ⃗β 1 + · · · + c i · ⃗β i + c j⃗β j + · · · + c n · ⃗β n↦→ c 1 · ⃗β 1 + · · · + c i · (⃗β i − k⃗β j ) + · · · + (kc i + c j ) · ⃗β j + · · · + c n · ⃗β n = ⃗v