How to verify that a basis is indeed a basis? (and Vector Spaces)

Question

How to verify that a basis is indeed a basis? Would you please walk me through the following three short problems? Thanks.

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Is basis basically is a set of linearly independent vectors that can construct every vector in a given space (subspace)? — user12488, Feb 12 '14 at 07:52
Would { (1,0), (i,0), (0,1), (0,i) } be one, since it deals with complex numbers? — user12488, Feb 12 '14 at 07:57
Related: http://math.stackexchange.com/questions/123448/what-is-the-standard-basis-for-fields-of-complex-numbers — Dahn, Feb 12 '14 at 08:01
If $\mathbb C$ is the set of scalars, then we would have $i\cdot (1,0)=(i,0)$. You don't need $(i,0)$. — Brian Fitzpatrick, Feb 12 '14 at 08:01
Oh! Thanks Brian, so does it mean for (a), I only need (1,0) and (0,1), but for part (b), I would need { 1, i }? So for (c) it would be { (1,0), (i,0), (0,1), (0,i) } ? Since for part c, scalars can only be reals — user12488, Feb 12 '14 at 08:02

score 2 · Accepted Answer · answered Feb 12 '14 at 08:05

Follow the definition of a basis; so, if $B=\{v_1,\dots,v_n\}$ is a basis, it will need to

Span the space. That is, for any $x\in V$, there exists $a_1,\dots,a_r\in\mathbb{C}$ such that $a_1v_1+\dots+a_rv_r=x$.
Have all its elements be linearly independent. That is, $a_1v_1+\dots+a_rv_r=0\implies a_1=\dots=a_r=0.$

Once you have checked the two conditions, you would have "verified" that it is indeed a basis.

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