Question

VUOL AS 80 CALDAS VO rool Graph VOOL Problen 4 Consider the liner transformation defined by T(1) = 1, T(x)=x-1 T(2²) = x²-ax T (2²) = x²-3x whose domain in P₂ the vector space of all polynomials of degree 3 or less. matrix representation of T with respect to the standard 1 tind busis A the (1.x, x standard x²) (2 compute the det (A) 3) ls T invertible 2 Explain it? 4 Compute the Rank and Nullity of A

Answer 1

DOMS Det consider linear trame formation T TC)=1 T00 - 744 Tox²) = x² Ton² = n° 3x² Domain of T is P3 A . standard martin representatim f. I with respect to the standard basis (1,0, 22, 23) NOW T(1) = a.lt ba tcx²4 dx3 = 1 aal b=o=d=0 T(1) z Intoxtox² tox3 = Pop - T A C ) T(4) 2 - anthontca ² t d 3 =) a=t, b=1 codzo T( 1 ) = -1H4 +0: 2 0 13 Also, T 1x2)= a.lt butca²taq3 = x²-22 920, 62-222!, do - T(x²) = o.lthax tix² tons NOW TCMB) = a.ltb.xtxeta a3 = a-3x² =) a=0, b=1,c20 d=-3 -(iv) T(43) = o.lt1.xtoon +(-3) x3 from (i), (i), (ii) and cru). 007 AO-21 OOO 1000-33

DOMS 2 det (A): - o of 0 0 0 ooo -3 21XL -2 1- ( 11 o 2 1 00-3 1x [(-3-0) –(-2) () +1 (0)] -(-1)(0) 21 x [-370 +038-0 2 -3 det A= -3 - P is invertible i det A t0 = Ã exist - f is onto transformation =) T is invertible 021 Too03 This is row-echlon form af A Tank (A)? no of non-zero rows in row-echlon form s rank(A) = 4 € = 4

DOM5 Page No. Date 3 rank nullity theorem Rank (A)+ Nullity of A = No of row's = 4 2) Naelbity 74 - 2- Rank (A) - 4.4 ( from ) 20 Hence Nullity (A) = 0