Determine whether the set {(2, 0, 1), (1, 1, 0), (0, 0, 1)} spans R3. Is this set linear independent?

Question

Determine whether the set {(2, 0, 1), (1, 1, 0), (0, 0, 1)} spans . Is this set linear independent?

Answer 1

R³ has dimension 3. So, a set of 3 vectors in R³ will span R³ as long as the set is linearly independent.

The vectors are linearly independent if the only real numbers a, b, c such that

a(2,0,1) + b(1,1,0) + c(0,0,1) = (0,0,0) are a = b = c = 0.

From the equation above, we have 2a + b = 0, b = 0 and a + c = 0.

It's easy to see that a, b, and c must all be zero.

So, the set is linearly independent and must span R³.