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A basis for a vector space is a set of vectors that are linearly independent and span the entire vector space. In other words, every vector in the vector space can be written as a linear combination of the basis vectors and no basis vector can be written as a linear combination of the other basis vectors.

A subspace of a vector space is a subset that is itself a vector space using the same operations as the original vector space. In this problem, we are given that \(W\) is a subspace of the vector space \(V\).

Let \(\{w_1, w_2, ..., w_n\}\) be a basis for \(W\). We need to show that this basis can be extended to a basis for \(V\). To do this, we will first show that the basis for \(W\) is linearly independent in \(V\). Since the basis vectors for \(W\) are linearly independent in \(W\), there are no coefficients \(c_1,c_2,...,c_n\) such that \(\sum_{i=1}^n c_iw_i=0\) other than \(c_i=0\) for all \(i\). This property holds in \(V\) as well since \(W\) is a subspace of \(V\). Therefore, the basis for \(W\) is linearly independent in \(V\).

Now we will construct a basis for \(V\) that includes the basis for \(W\). We start by adding vectors from \(V\) to the basis for \(W\). If the new vector that we add is linearly independent from the basis vectors for \(W\), we add it to our list. We continue this process until we have a basis for \(V\) that includes the basis for \(W\). If \(V\) is a finite-dimensional vector space, we can perform the above process until we have a basis for \(V\) that includes the basis for \(W\). If \(V\) is an infinite-dimensional vector space, we can still perform the above process and construct a basis for \(V\) that includes the basis for \(W\), but we will use a process called Zorn's lemma to prove that such a basis exists. We won't go into details of Zorn's lemma in this solution.

We have shown that any basis for the subspace \(W\) is a set of linearly independent vectors in the vector space \(V\) and therefore can be extended to form a basis for \(V\). This holds for both finite-dimensional and infinite-dimensional vector spaces.