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Understanding vector spaces is fundamental for grasping concepts involving linear transformations. A vector space is a collection of vectors where vector addition and scalar multiplication are defined. These spaces are equipped with certain rules, such as the existence of a zero vector and the ability to perform linear operations.
A vector space can be formed by a set of vectors, like \([-1, 2]\) and \[2, 1]\], which can be combined linearly to produce any vector in that space, such as \[0, 10]\].
This idea becomes crucial when dealing with linear transformations, as it allows us to express new vectors in terms of known ones. The set of all linear combinations of a given set of vectors forms a vector subspace, illustrating the powerful concept of span in vector spaces.

Linearity is a fundamental property of linear transformations and is key to solving the problem given in the exercise. Linear transformations preserve vector addition and scalar multiplication, which mathematically translates to: \[ T(a \, \mathbf{u} + b \, \mathbf{v}) = a \, T(\mathbf{u}) + b \, T(\mathbf{v}) \].
This property allows us to decompose complex vectors into simpler components whose transformations are known.
The linear transformation doesn't distort the vector space's structure, ensuring that parallelism and origin point remain consistent across transformations. For example, in our problem, knowing the transformations of \([-1, 2]\) and \[2, 1]\], we could confidently deduce the transformation for \[0, 10]\] using their linear combination.

A linear combination involves expressing a vector as the sum of scalar multiples of other vectors. In the exercise, the methodology revolved around expressing the vector \[0, 10]\] as a linear combination of \([-1, 2]\) and \[2, 1]\].
Mathematically, this expression took the form: \[ a[-1, 2] + b[2, 1] = [0, 10] \]. By solving this equation, we found the values of \("a"\) and \("b"\) to be 4 and 2, respectively.
Linear combinations allow for the simplification and breakdown of complex vectors into known vectors within the vector space, facilitating easier transformations and calculations. This process is central to vector space theory and problem-solving in linear algebra.

To understand transformations better, we can use matrices. A linear transformation can be represented by a matrix, which acts on a vector to produce another vector. This representation is very powerful as it simplifies computations and provides a clear framework for understanding complex transformations.
In matrix form, transformations utilize rows and columns to define how exactly the vectors are altered. If we had \[T(\mathbf{x}) = A\mathbf{x}\], the matrix \(A\) would fully encode the transformation \(T\).
Though not explicitly used in this exercise, knowing the matrix representation of a transformation allows us to capture all aspects of its linearity and quickly determine how different vectors are mapped. This is especially helpful for visualizing and computing transformations on a larger scale.