Linear Algebra

Section V. Change of Basis 241All 2×2 matrices:⋆(0 0)0 0⋆(1 0)0 1⋆(1 0)0 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, but the other twoclasses each have 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 it is expressed in this way: if thebases are B = 〈 ⃗ β 1 , . . . , ⃗ β n 〉 and D = 〈 ⃗ δ 1 , . . . , ⃗ δ m 〉 then the map sendsc 1⃗ β1 + · · · + c k⃗ βk + c k+1⃗ βk+1 + · · · + c n⃗ βn↦−→ c 1⃗ δ1 + · · · + c k⃗ δk + ⃗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+1↦→c k 0⎜ ⎟ ⎜ ⎟⎝ . ⎠ ⎝ . ⎠c n0BOf course, “understanding” a map expressed in this way requires that we understandthe relationship between B and D. However, despite that difficulty,this is a good classification of linear maps.DExerciseš 2.10 Decide ( if these ) ( matrices are ) matrix equivalent.1 3 0 2 2 1(a),2 3 0 0 5 −1( ) ( )0 3 4 0(b) ,1 1 0 5( ) ( )1 3 1 3(c) ,2 6 2 −6̌ 2.11 Find the canonical representative of the matrix-equivalence class of each matrix.