Question

13. (Linear Algebra Review) Recall that a function T:R" + R" is one-to-one if for each b ER", there is at most one x ER" such that T(x) = b. Let T be a linear transformation given by a matrix A. If T is one-to-one, then what can be said about the columns of A? What can be said about the determinant of A?

Answer 1

• 
  1 13. Given that T1RW_Rh such that T(X) = AX Where A=(a;;) new w Note that as I is one to one, Hence wsm [ This is because or TRW), is a subspau of and din TIRA) = n. is ze... ews of Jeh Now we consider the standard basis ze... wg es Then I(e) = Ae;=ith column of A = 0; ug e = [ a , aeg, ami] We claim that C's i=J... W are Lincaily Sudependent. fet x;, i=1... WER such that Š x 4 = 0 *Ž*Tei) = 0 → Ž T60;lt) =0 => Taxei) = 0 As T is one to one Å xierzo Aspeisian is a basis of Rh .. Hence x;=0 fiar... m a 2) pes are lineally independent. Hence, columnes of A are lineaal ing of A are Lineally Independent. naut of A is considered, then nim when determinant of A is considere Then diu MIRW) die am → T(IR") = RW ū T į abijective map. u inverse Hence det A fo. ective map. in inverre of Texists, ü Ä exols.