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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?

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  • $\begingroup$ Note that any set that contains the zero vector must be linearly dependent. $\endgroup$ – Théophile Feb 3 '16 at 3:10
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(ii) contains only 3 linearly independent vectors, so cannot span a 4-dimensional space.

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

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  • $\begingroup$ Got it, thanks! $\endgroup$ – MilTom Feb 3 '16 at 18:54
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(i) contains 5 vectors, so they can certainly not be linearly independent. Similarly, (iv) only contains 3 vectors, so certainly cannot span a 4-d space.

(iii) notice that the fourth vector is a multiple of the second one

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