Question

2. (8 marks) Let dim V = dim W = n. Let T:V → W be a linear transformation of rank k. Show that there are ordered bases B of V and D of W such that MdB(T) is a matrix that contains exactly k entries that are ls (with the rest being Os).

I know theorem states that rank T = Mdb(T). but what is next?
Thank you very much

Answer 1

Given that dim v= dim Wah and TEV+W be a linear trans Formation of .. Rankk. > dimension of Range space of T = Rank T * -> dimension of Range space of Tak . But we know that .. if BEL VI, V2,-- Um} Are basis of V Then S=(ui), T(V2)-- , TCVm} ☺ is span of Range space of W. But By 6, dimension of Range space ok , out of m vectors in s there there are only k vectors which are lineany independent, Suppose that vectes B = { TCVO I (Vma), -- Tlume)? Cs

. Extend Basis Bi of Range space of I to Basis of W by adding (n-k). lineany independent vectors vectors, which are outside span of Bi > ] = { reunu), tevnu), -- TCUDE), , 6027. Wn*} Ć Basis of bodo . Now take matrix , OF T wst. csdered Basis B OF V and B of la TTTIILIKUITU That matrix will contain exactly K entries that are I. and Rest of the other are teso (o).