# Why are these not bases in $\mathbb{R}^4$?

I know that bases vectors must span and be linearly independent.

The (i) is not bases because the last vector contains $\pi$. The (iii) is not bases because they are not linearly independent. The (iv) is not bases because there are only 3 vectors here and bases for $\mathbb{R}^4$ must contain at least 4 vectors. The (v) is not bases because they do not span. What about (ii), is it a basis?

• Note that any set that contains the zero vector must be linearly dependent. – Théophile Feb 3 '16 at 3:10

Also, your answer for (i) is wrong: the $\pi$ is irrelevant. Count the vectors again. Can they be linearly independent?