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Linear independence is a fundamental concept in the study of vector spaces. It tells us whether a set of vectors can be expressed in terms of each other or not. To determine if the set \( \mathcal{B} = \{v_1+v_2, v_2+v_3, v_2\} \) is linearly independent, we look for constants \(a, b,\) and \(c\) such that the linear combination \(a(v_1+v_2) + b(v_2+v_3) + c v_2 = 0\) holds only when all constants are zero.

Substituting these vectors, we obtain \( a v_1 + (a + b + c) v_2 + b v_3 = 0\). Since \( \mathcal{A} \), the original basis, is also a basis of \(V\), it implies that the only solution is \( a = 0, b = 0, \) and \( c = 0\). If this is true, then the vectors in \(\mathcal{B}\) do not express any vector as a nontrivial combination of themselves, maintaining their independence.

This property ensures that the set \( \mathcal{B} \) is linearly independent, which is one of the requirements for \(\mathcal{B}\) to be a vector space basis.

A basis in a vector space is a set of vectors that are linearly independent and span the entire space. In simpler terms, you can think of a basis as a minimal set of vectors needed to construct every possible vector in the space. For the set \( \mathcal{B} = \{v_1+v_2, v_2+v_3, v_2\} \), establishing a basis involves two critical checks: linear independence and span.

We've already established linear independence in the previous step. Now, let's look at spanning. The set \( \mathcal{B} \) must cover the whole space \(V\). We confirm this by expressing any vector \(v = x v_1 + y v_2 + z v_3\) as \(a(v_1+v_2) + b(v_2+v_3) + c v_2\). Solving for \(a, b,\) and \(c\), gives us a mechanism to express all possibilities in \(V\), confirming the set \(\mathcal{B}\) spans \(V\).

Since \( \mathcal{B} \) is both linearly independent and spans the vector space, it is a valid basis of \(V\). Understanding this helps conceptualize how vector spaces are structured and how changes in bases affect representation.

Matrix representation provides a structured way to understand bilinear forms and their transformations between different bases in a vector space. Starting with the matrix \( M_{\mathcal{A}}(s) \) associated with the original basis \(\mathcal{A}\), the challenge is to find a corresponding matrix \( M_{\mathcal{B}}(s) \) for the new basis \( \mathcal{B} \).

The transformation involves a special matrix known as the transition matrix, \(P\), which maps vectors from the original basis to the new basis. The transition matrix for \( \mathcal{B} \) expressed in terms of \( \mathcal{A} \) is: \[ P = \begin{pmatrix} 1 & 0 & 0 \ 1 & 1 & 1 \ 0 & 1 & 0 \end{pmatrix}. \]By calculating \( M_{\mathcal{B}}(s) = P^T M_{\mathcal{A}}(s) P\), we can transform the bilinear form from its original basis to the new one.

The resulting matrix \(M_{\mathcal{B}}(s)\) encapsulates all the transformations of the bilinear form \(s\) as it is now represented in terms of the new basis \( \mathcal{B} \). Remember, matrix transformations are essential for translating between different frames of a vector space, which can significantly simplify problem-solving or provide deeper insights.