Proof that the all bases of a subspace V consist of the same number of vectors?

Question

My textbook explains the proof, which I don't understand:

"Consider two bases $v_1...v_p$ and $w_1...w_q$ of V. Since the vectors $v$ are linearly independent and the vectors $w$ span V..."

How exactly does $w$ span V?

The book then says the same for vectors $v$, that $v$ spans V and hence $p=q$, but I don't really understand how you can assume that given two sets of vectors that are basis, one set must span V.

Answer 1

A subset $B$ of a vector space $V$ is a basis for $V$ if it spans $V$ and is linearly independent.

If $\{v_i\}_1^p$ and $\{w_k\}_1^q$ are bases of $V$, then:

1. Both $\{v_i\}$ and $\{w_k\}$ span $V$,

2. Both $\{v_i\}$ and $\{w_k\}$ are linearly independent sets.

Answer 2

By definition, a basis for a (say real) vector space $V$ consists of a family of vectors $(v_1,\ldots,v_q)$ that are $\mathbb{R}$-linearly independant and that span the vector space V. The fact that they span $V$ means any vector $v\in V$ can be written as a linear sum of elements of the basis, which means there exists reals $a_1,\ldots,a_q$ scalars such that $v=a_1v_1+\ldots + a_qv_q$.