In Axler's Linear Algebra Done Right there is the posit that every linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis of the vector space. This seems fine, intuitively, but I don't understand his proof. He states:
Suppose a linear independent list, $u_1$ , ... , $u_m$ exists in vector space V. V also has a basis $w_1$ , ... , $w_m$.
1) If one creates a list $u_1$ , ... , $u_m$ , $w_1$ , ... , $w_m$, that means you now have another list that spans V.
2) Further, he says that if you remove any $w_j$ elements within the new list, you're able to create a basis as long as you don't remove any $u_j$ elements, as that original list of $u_j$ elements is linearly independent.
My questions are regarding 1) and 2). For 1): How is this new list a span of V? And for 2): Why does he emphasize that you need to remove $w_j$ elements and not $u_j$? Since the $w_j$ elements were originally from a basis, aren't they linearly independent, also?*