Question

6 and 7!!!

6. Given the points A - (0,0), B = (5,1),C - (2,6) on the plane. Use a determinant to find the area of the triangle ABC. 7. Let T:V - W be a linear transformation, and let 01,03,...,U, be vectors in V. Suppose that T(u), 7(v2),...,T(v.) are linearly independent. Show that 01,03,.., are linearly independent

Answer 1

Area of the triangle whose vertiees are (24,9.) , (az, ), (13, 73) 4 8 is =+1 2 area will be always to 22 Yz Az Y3 So, Area of the thiagle ABC A = (0,0), B = (5,1), C = (2,6) where O is + 1 1) 2 5 2 6 U I+ (1-6)-0(5-2)+1 (30-2)] 2 14 = 늘 x 28 (7) T:V >W be linean tras formation let {o,,ų, ..., on? be the vectons from V such that {+(0), 7(0) , - .,T(on)] cine lineanly independent vectors of w have to prove so,, Q, ... On} linearly independent vectors of are we also linearly CS Scanned with CamScanner

zero vectol . we can prove ve we are Sinee Then T is So, let e, o,+CO₂ + ten Un= Ov Clean of v G = ez = en = done. linen tras formation, → Tlau + C Ut en cen) = T(Q) (zero → TlQ41)+ T(220) + Tlen Un) = Owl vector w → CT(U) + C, TUL) + + en Tcun) = Ow Since T(",), T(U). --, T(Un) is linenly independent, c, = e = ---=en=0 on] lineonly independet vectons of v v Henee {0,,U2, .. 7 CS Scanned with CamScanner