Why does linearly independent spanning set imply minimal spanning set for a vector space? Suppose β is a linearly independent spanning set of some vector space V. Why must it be the minimal spanning set?

In other words, why can there not be two linearly independent spanning sets of a vector space V with different sizes?
Suppose $v_1,\ldots,v_n$ and $w_1,\ldots,w_m$ are bases for $V$ where $n&lt;m$.

By writing the $w_i$ in terms of the $v_j$, you create a system $w_i=\sum_j \alpha_{ij}v_j $.

In terms of coordinates, this means that the vector $w_i$ has coordinates $[\alpha_{i1},\alpha_{i2},\ldots,\alpha_{in}]$ in the basis of $v_i$'s. In other words, you have $m$ length $n$ vectors that you believe are linearly independent (since the $w_i$ are linearly independent.) You can put these in a matrix, and keep in mind you believe the rows to be linearly independent.

By using elementary row operations, though, you can put this matrix into row-echelon form, where it must have all zeros in the last row. This says that the last row is a linear combination of the rows above it, and that means that the $w_i$ weren't linearly independent after all. This contradiction shows why the assumption in the first line can never occur.
Just a quick explanation from dimension theorem of vector space. 

http://en.wikipedia.org/wiki/Dimension_theorem_for_vector_spaces

Suppose you have $v_1$ and $v_2$; they are linearly independent and span $V$. Then they form the basis for $V$. As such the dimension of $V$ is $2$ and according to the dimension theorem, all set of linearly independent vectors that span $V$must be of size $2$ as well.