a Check if the set V 324 243 013 is formed by Linear indepen

(a) Check if the set V= ((3,2,4), (2,4,3), (0,1,3)) is formed by Linear independent equations. (b) Check if the set V span R^3. (c) Is the set V a basis for R^3? Why?

Solution

In order to check linear dependence of vectors, we need to check that

av1 + bv2 + cv3 = 0 only if all a,b and c are all zeroes

a(3,2,4) + b(2,4,3) + c(0,1,3) = 0

3a + 2b = 0

2a + 4b + c = 0

4a + 3b + 3c = 0

b = -3a/2 from first equatio

2a + 4(-3a/2) + c = 0 => c =4a

4a + 3(-3a/2) + 12a = 0

=> a = 0, since b = -3a/2 = 0 and c = 4a = 0

Hence all a,b and c are equal to zero implies that all these vectors are linearly independent

b) To check if the set V span R^3

since all the vectors in V belongs to the region R^3 and all of them are linearly independent, hence they span the vector space of R^3

c) Is the set V a basis of R^3

For R^n, we need minimum of n vectors to form basis which span the complete space

In the case of R^3, all the vectors are in the region R^3 and they are equal to 3

Hence they span the vector space R^3 and form the basis for the same