Linear Algebra

Section V. Change of Basis 245Finally, combine the left-multipliers together as P and the right-multiplierstogether as Q to get the PHQ equation.⎛⎞⎛⎞ ⎛1 −1 0⎜⎟ ⎜1 2 1 −1⎞ 1 0 −2 0 ⎛⎟0 0 1 0⎝ 0 1 0⎠⎝0 0 1 −1⎠⎜⎟⎝0 1 0 1⎠ = ⎜1 0 0 0⎞⎟⎝0 1 0 0⎠−2 0 1 2 4 2 −20 0 0 00 0 0 12.8 Corollary Two same-sized matrices are matrix equivalent if and only if theyhave the same rank.That is, the matrix equivalence classes are characterized by rank.Proof Two same-sized matrices with the same rank are equivalent to the sameblock partial-identity matrix.QED2.9 Example The 2×2 matrices have only three possible ranks: zero, one, or two.Thus there are three matrix-equivalence classes.All 2×2 matrices:⋆( ) 0 00 0⋆( ) 1 00 1⋆( ) 1 00 0Three equivalenceclassesEach class consists of all of the 2×2 matrices with the same rank. There is onlyone rank zero matrix so that class has only one member. The other two classeshave infinitely many members.In this subsection we have seen how to change the representation of a mapwith respect to a first pair of bases to one with respect to a second pair. Thatled to a definition describing when matrices are equivalent in this way. Finallywe noted that, with the proper choice of (possibly different) starting and endingbases, any map can be represented in block partial-identity form.One of the nice things about this representation is that, in some sense, wecan completely understand the map when we express it in this way: if the basesare B = 〈⃗β 1 , . . . , ⃗β n 〉 and D = 〈⃗δ 1 , . . . ,⃗δ m 〉 then the map sendsc 1⃗β 1 + · · · + c k⃗β k + c k+1⃗β k+1 + · · · + c n⃗β n ↦−→ c 1⃗δ 1 + · · · + c k⃗δ k +⃗0 + · · · +⃗0where k is the map’s rank. Thus, we can understand any linear map as a kindof projection.⎛ ⎞ ⎛ ⎞c 1 c 1 ..c kc k ↦→c k+10⎜ ⎟ ⎜ ⎟⎝ . ⎠ ⎝ . ⎠0c nBOf course, “understanding” a map expressed in this way requires that we understandthe relationship between B and D. Nonetheless, this is a good classificationof linear maps.D