Linear Algebra

Section III. Computing Linear Maps 201a matrix of interest does not represent a linear map on some pair of spaces ofinterest. (In practice, when we are working with a matrix but no spaces orbases have been specified, we will often take the domain and codomain to be R nand R m and use the standard bases. In this case, because the representation istransparent — the representation with respect to the standard basis of ⃗v is ⃗v —the column space of the matrix equals the range of the map. Consequently, thecolumn space of H is often denoted by R(H).)With the theorem, we have characterized linear maps as those maps that actin this matrix way. Each linear map is described by a matrix and each matrixdescribes a linear map. We finish this section by illustrating how a matrix canbe used to tell things about its maps.2.3 Theorem The rank of a matrix equals the rank of any map that itrepresents.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 rangespace{h(⃗v) ∣ ∣ ⃗v ∈ V } = {h(c 1⃗ β1 + · · · + c n⃗ βn ) ∣ ∣ c 1 , . . . , c n ∈ R}= {c 1 h( ⃗ β 1 ) + · · · + c n h( ⃗ β n ) ∣ ∣ c 1 , . . . , c n ∈ R}is the span [{h( β ⃗ 1 ), . . . , h( β ⃗ n )}]. The rank of h is the dimension of this rangespace.The rank of the matrix is its column rank (or its row rank; the two areequal). This is the dimension of the column space of the matrix, which is thespan of the set of column vectors [{Rep D (h( β ⃗ 1 )), . . . , Rep D (h( β ⃗ n ))}].To see that the two spans have the same dimension, recall that a representationwith respect to a basis gives an isomorphism Rep D : W → R m . Underthis isomorphism, there is a linear relationship among members of the rangespaceif and only if the same relationship holds in the column space, e.g, ⃗0 =c 1 h( β ⃗ 1 ) + · · · + c n h( β ⃗ n ) if and only if ⃗0 = c 1 Rep D (h( β ⃗ 1 )) + · · · + c n Rep D (h( β ⃗ n )).Hence, a subset of the rangespace is linearly independent if and only if the correspondingsubset of the column space is linearly independent. This means thatthe size of the largest linearly independent subset of the rangespace equals thesize of the largest linearly independent subset of the column space, and so thetwo spaces have the same dimension.QED2.4 Example Any map represented by⎛ ⎞1 2 2⎜1 2 1⎟⎝0 0 3⎠0 0 2must, by definition, be from a three-dimensional domain to a four-dimensionalcodomain. In addition, because the rank of this matrix is two (we can spot this