Q:

Dimension of $\Lambda ^k (V)$

In Calculus on Manifolds by Spivak theorem 4-5 is as follows:

The set of all $$ \varphi _{i_1} \wedge \dots \wedge \varphi _{i_k} \quad 1\leq i_1<i_2<\dots<i_k\leq n$$
  is the basis for $\Lambda ^k (V)$, which therefore has dimension $\binom{n}{k}$.

Why do we have the condition $ 1\leq i_1<i_2<\dots<i_k\leq n$ (strict order) ?   

A:

Because $\varphi_{i_1} \wedge \dots \wedge \varphi_{i_k}$ and $\varphi_{i_{j_1}} \wedge \dots \wedge \varphi_{i_{j_k}}$, where $(i_{j_1},\dots,i_{j_k})$ is a permutation of $(i_1,\dots,i_k)$, are linearly dependent.  Further, since $\wedge$ is skew-symmetric, whenever two of the indices coincide your element is actually $0$.

