Homework SourseWhich of he following are a basis of R3 Justify your answer
Which of he following are a basis of R^3? Justify your answer with the appropriate steps.
(a) (1,2,3), (3,2,1), (1,1,1), (1,0,1)
(b) (0,1,0), (1,0,0), (1,0,1)
(c) (2,2,0), (1,1,1) (0,0-1)
Which of he following are a basis of R^3? Justify your answer with the appropriate steps.
(a) (1,2,3), (3,2,1), (1,1,1), (1,0,1)
(b) (0,1,0), (1,0,0), (1,0,1)
(c) (2,2,0), (1,1,1) (0,0-1)
(a) (1,2,3), (3,2,1), (1,1,1), (1,0,1)
(b) (0,1,0), (1,0,0), (1,0,1)
(c) (2,2,0), (1,1,1) (0,0-1)
Solution
a)
R3 has dimension 3 so basis should have 3 vectors
But this set has 4 vectors so it is not a basis
b)
(1,0,1)-(1,0,0)=(0,0,1)
Hence all standard basis vectors of R3 are in span of this set
and this set has 3 vectors so this set is the basis for R3
c)
(0,0,-1)=(2,2,0)/2-(1,1,1)=(0,0,-1)
Hence given set of three vectors is linearly dependent but a basis is linearly independent set
So this set is not a basis
