16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-9andT (αx) = αT (x)One can easily see that maps from R n to R m defined using multiplicationof vectors x by m × n matrices A as above are linear maps.Conversely, it is relatively easy to prove that every linear map T : R n →R m is defined by multiplication by some m × n matrix.To be actually correct in this case, it is easier to ignore our convention ofwriting vectors as either row or column vectors, and to stick to row vectors.That is, we write x = (x 1 , x 2 , . . . , x n ) as a 1 × n matrix.Then, we have the followingProposition 4.2 If T : R n → R m is a linear map, then there is an n × mmatrix A such T (x) = x · A for all x ∈ R n .Proof. Let {e 1 , e 2 , . . . , e n } denote the standard unit vectors in R n , andlet {f 1 , f 2 , . . . , f m } denote the standard unit vectors in R m .Observe that writing a vector x ∈ R n as x = (x 1 , x 2 , . . . , x n ) amounts tosame asn∑x = x i e i .i=1asSimilarly, if y ∈ R m is written as y = (y 1 , . . . , y m ), then this is the samem∑y = y j f j .j=1Let T : R n → R m be a linear map.By linearity, we haven∑n∑T (x) = T ( x i e i ) = x i T (e i ) (5)i=1i=1Now, each T (e i ) is a vector in R m , so there are real numbers a ij suchthatm∑T (e i ) = a ij f jj=1