Linear Algebra

Section III. Computing Linear Maps 2032.3 Example Even if the domain and codomain are the same, the map that thematrix represents depends on the bases that we choose. If( )( ) ( )( ) ( )1 01 00 1H = , B 1 = D 1 = 〈 , 〉, and B 2 = D 2 = 〈 , 〉,0 00 11 0then h 1 : R 2 → R 2 represented by H with respect to B 1 , D 1 maps( ) ( )) )c 1 c 1=↦→c 2 c 2B 1B 2(c 10D 1=D 2=(c 10while h 2 : R 2 → R 2 represented by H with respect to B 2 , D 2 is this map.( ) ( ) ( ) ( )c 1 c 2c 2 0=↦→c 2 c 1 0 c 2These are different functions. The first is projection onto the x-axis, while thesecond is projection onto the y-axis.This result means that we can, when convenient, work solely with matrices,just doing the computations without having to worry whether a matrix of interestrepresents a linear map on some pair of spaces. When we are working with amatrix but we do not have particular spaces or bases in mind then we oftentake the domain and codomain to be R n and R m and use the standard bases.This is convenient because with the standard bases vector representation istransparent — the representation of ⃗v is ⃗v. (In this case the column space of thematrix equals the range of the map and consequently the column space of H isoften denoted by R(H).)We finish this section by illustrating how a matrix can give us informationabout the associated maps.2.4 Theorem The rank of a matrix equals the rank of any map that it represents.Proof Suppose that the matrix H is m×n. Fix domain and codomain spacesV and W of dimension n and m with bases B = 〈⃗β 1 , . . . , ⃗β n 〉 and D. Then Hrepresents some linear map h between those spaces with respect to these baseswhose range space{h(⃗v) ∣ ∣ ⃗v ∈ V } = {h(c1 ⃗β 1 + · · · + c n⃗β n ) ∣ ∣ c1 , . . . , c n ∈ R}= {c 1 h(⃗β 1 ) + · · · + c n h(⃗β n ) ∣ ∣ c1 , . . . , c n ∈ R}is the span [{h(⃗β 1 ), . . . , h(⃗β n )}]. The rank of the map h is the dimension of thisrange space.The rank of the matrix is the dimension of its column space, the span of theset of its columns [{Rep D (h(⃗β 1 )), . . . , Rep D (h(⃗β n ))}].To see that the two spans have the same dimension, recall from the proofof Lemma I.2.5 that if we fix a basis then representation with respect to that