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To understand the process of finding the transformation of a vector in a different space, we first need to talk about basis vectors. Imagine a coordinate system as a unique type of map. In three-dimensional space, any point can be described using three specific directions. These directions are known as basis vectors. For example, the vectors \((1,1,1)\), \((0,-1,2)\), and \((1,0,1)\) can serve as a set of basis vectors for \(\mathbb{R}^3\).

• Each basis vector is independent of the others.
• Together, they can represent any vector in the space by taking combinations of these directions.

When solving problems involving linear transformations, establishing a set of basis vectors helps us navigate how each vector in space is transformed. This becomes the foundation for expressing vectors as a linear combination, the next critical concept.

Once we have our basis vectors, we can express any vector as a linear combination of these basis vectors. Simply put, a linear combination involves adding together multiple vectors, each multiplied by a scalar (a constant number). For example, suppose you have a vector \((0,2,-1)\) that you want to express using the basis vectors \((1,1,1), (0,-1,2), (1,0,1)\). You could find scalars \(a\), \(b\), and \(c\) such that:
\[(0, 2, -1) = a \times (1, 1, 1) + b \times (0, -1, 2) + c \times (1, 0, 1)\]

• This means identifying how much of each basis vector contributes to the vector \((0,2,-1)\).
• In our solution, these turned out to be \(a=1\), \(b=-1\), and \(c=-1\).

This method allows us to rewrite \((0,2,-1)\) in terms of known, mapped transformations of the basis vectors, making it easier to apply linear transformations.

Now, let's delve into how vector transformation works with linear transformations. A linear transformation is a function that maps vectors to other vectors in a consistent manner, meaning the transformation respects vector addition and scalar multiplication. For our exercise, the transformation is represented by \(T((x,y,z))\), where we aim to find the image of a given vector under this transformation.
The rule of thumb with transformations is that transforming a combination of vectors yields a combination of their transformed images. If you have equations for \(T(x)\), \(T(y)\), and \(T(z)\), you can then find \(T(v)\), where \(v\) is a vector expressed as a linear combination of the basis vectors:
\[T(v) = a \times T(x) + b \times T(y) + c \times T(z)\]

• Each transformation \(T(x)\), \(T(y)\), and \(T(z)\) reflects how each basis vector is transformed.
• In our case, this culminates in determining \(T(0,2,-1)\) by substituting our values of \(a\), \(b\), and \(c\).

Thus linear transformations help in predicting and determining where any vector ends up after the transformation by mapping its components separately and combining them as per their linear combination coefficients.