Question

Can the set {(-1.0.0). (0.0,0), (0.0,3)} be used to span R3? Explain.

Answer 1

Given the set of vectors {u, v, w}:

$\begin{gathered} u=(-1,0,0) \\ v=(0,0,0) \\ w=(0,0,3) \end{gathered}$

This set is said to be linearly independent if there is a non-trivial solution to the equation:

$C_1u+C_2v+C_3w=(0,0,0)$

Where C₁, C₂, and C₃ are real constants. In this case, we see that if C₁ and C₃ are 0, we will have:

$C_2v=(0,0,0)$

But v is already the null vector (0, 0, 0), so it doesn't matter the value of C₂ this equation will always be true. Then, we found a non-trivial solution, hence the system is linearly dependent.

If it is linearly dependent, it can not be used to span R³