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When dealing with linear transformations, a change of basis can sometimes be a necessary step. The basis of a vector space is essentially a set of vectors that is linearly independent and spans the space. Any vector in this space can be written uniquely as a linear combination of the basis vectors.

In the exercise, we initially have our vector \( \mathbf{v} \) expressed in terms of the basis \( B' = \{(-1,1), (1,2)\} \). To work with it in the context provided, we will transform it to another basis \( B = \{(1,-1), (-2,1)\} \). The change of basis involves finding this new representation, which comes from understanding how the vector \( \mathbf{v} \) can be expressed in terms of different sets of vectors.

By changing the basis, we are essentially translating the vector into a different 'language' or perspective, which can simplify computations or align with other parts of our problem setup. This process of basis change can be crucial in contexts like simplifying a system of equations or finding a more intuitive geometric interpretation.

Coordinate vectors provide a way to express any vector uniquely in terms of the basis vectors of the space. When we talk about a coordinate vector, such as \([\mathbf{v}]_{B}\), we are referring to the specific list of numbers that describes how our vector \( \mathbf{v} \) is constructed using the basis \( B \). Each component of this coordinate vector corresponds to a scaling factor for a particular basis vector.

For example, in the solution, we find \([\mathbf{v}]_{B} = [-3, 1]^T\). What this means is that the vector \( \mathbf{v} \) is made up of -3 times the vector \((1, -1)\) and 1 times the vector \((-2, 1)\). These coordinate vectors are essential in linear algebra because they allow us to manipulate and transform vectors easily, especially across different bases.

It is important to distinguish between the vector itself and its representation in terms of a basis—much like how the same physical location might have different longitude and latitude depending on the coordinate system used.

A system of linear equations involves multiple linear relationships between variables. In the exercise, after determining the vector \( \mathbf{v} \) in the standard basis, we constructed a system to convert to a new basis \( B \). This system was determined as:

• \(-5 = x - 2y\)
• \(-7 = -x + y\)

These equations arise directly from combining the basis change with how the original vector \( \mathbf{v} \) is structured in the standard basis. Solving such systems allows us to find specific coefficients—\(x\) and \(y\)—that represent our vector in another basis.

Solving systems of linear equations can be done using various methods, such as substitution or elimination. The best method often depends on the complexity and nature of the problem. In this case, obtaining \(x = -3\) and \(y = 1\) gives us the coordinate vector \([\mathbf{v}]_{B}\). Understanding these systems equips us with the skills to navigate transformations and changes of dimensional views in linear algebra.