Question

Linear Algebra and Its Applications (5th Edition), Chapter 4.2, 4.3 - Col(A) & Nul(A)

5 1. meä [--3-- 6.9 min side of women [8] [7 8. Let vi = -4,72 = -6, -6 . Determine whether or not {01, U2, U3} is a basis R3. If the set 4 10 8 forms a basis, explain why. If the set does not form a basis, explain why not. Be sure to cite appropriate theorems in your explanations.

Answer 1

+ (3 c croit o when i us relation consider the a, C₂, C₃ ER. - -09-18 This giver 50, +8C2 + 73 = 0 -46-602-669 = 0 + 46+6C2 +6C3=0 40 + 10 +803 =0 This is a homogeneous system of equalions in a c2,3 The coefficient determinant 5 8 7 4 6 = 5 (48-60)-8/32-24) 4 +7 (40-24) 10 8 = -60-64 +112 .- =-12 +0 By cramer's role, there exists a unique solution and the solution is = =2= 0. This proves that the set 40, 52, 537 is linearly independent Since IR² is a vector space of dimension 3 and 20, 52, 53) is a linearly independent set containing 3 vectors of 1R², 40, 32, 3 is a basis of 1R3 Note: If v. be a vector space of dimension over a field F. Then any linearly independent Set of n vectors of v is a basis of v