When it comes to proving $\beta$ is linearly independent, I understand the steps until it writes
$"$then $\sum_{i =1}^i d_i*u_i+ \sum_{k=1}^n -c_k*w_k=0$, so $c_k=0$ for all k.$"$
How does it claim $c_k=0$? Is it by uniqueness of scalars so that the only possible way for the equation to equal $0$ is letting all scalars equal $0$?
The set $\{\mathbf{u}_1,\ldots,\mathbf{u}_l,\mathbf{w}_1,\ldots,\mathbf{w}_n\}$ is, by definition of the $w_i$, a basis for $W_2$, so in particular is linear independent. By definition, that means that the only coefficients satisfying $$\sum\limits_{i=1}^l d_i \mathbf{u}_i + \sum\limits_{j=1}^k c_j\mathbf{w_j} = \mathbf{0}$$ are $d_i = c_j = 0$ for all $i,j$.