Prove that the set S= {[1, 2, 1]. [1, 0, 2]. [1, 1, 0]} spans R^3

Answer 1

lR³ has dimension 3, so any set of 3 linearly independent vectors in lR³ is a spanning set.

So, we need to show that the given vectors are linearly independent.

If a(1, 2, 1) + b(1, 0 2) + c(1, 1, 0) = (0, 0, 0), then we must show that a = b = c = 0.

We have:

a + b + c = 0

2a + c = 0

a + 2b = 0

c = -2a and b = -(1/2)a

So, a - (1/2)a - 2a = 0

Therefore, -(3/2)a = 0

So, a = 0, b = -(1/2)a = 0, and c = -2a = 0

Thus, the given vectors are linearly independent and span lR³.