Question

SOLVE BOTH 6 and 7

6. Given the points A - (0,0), B = (5,1), (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, 02, ..., Urbe vectors in V. Suppose that T(u), 7(2),...,T(un) are linearly independent. Show that 01, 02,..., Un are linearly independent.

Answer 1

Solution:

for 26th) question: Given that: The points A = (0,0) - COC, y.) B =(5,1) -638242) C =(2,6) Ocgry3) To find Area of C(2,6) A ABC A زهره B (5) * By determinant Method 21 yil Area of A ABC- 1 xq Y x3 Y jot o O 5:1 2 6 => [0C1-6)-0(5-2) +1(30-2)] ->Ź (28] lly unit Therefore, Area of the triangle ABC = 14 units

for Cath) Question:- Given thati- Let T: VW bea linear transformation Let ving... Vobe Vectors in v. Here, * Since TCVI), TCV2),... TCVN) are Linearly independent, so there exist ocalars cu ce,...cn such that C17 (VI) + C2T (Vg)+...+ sot CVO) =0 implieb, alacq: * Now a5 Tiba dinear transformation, then we have, CiT (V.) + C T Cve) +...+ Cnt.CVN) => taivit G Vot-otcom) =cn=0 Now as a=62² cn=o then civitqV₂ + & Covno * This implies that the set {\lla... Vn} is Linearly independenti {proved