91Ó°ÊÓ

In linear algebra, basis vectors are fundamental to understanding vector spaces and transformations. A basis is a set of vectors that, through linear combinations, can represent any vector in the space. For a two-dimensional space like \( \mathbb{R}^2 \), the standard basis vectors are \( \begin{bmatrix} 1 \ 0 \end{bmatrix} \) and \( \begin{bmatrix} 0 \ 1 \end{bmatrix} \). These vectors help us uniquely express any vector in that space.

In our exercise, the transformation \( T \) is defined by its action on these standard basis vectors. Knowing how each basis vector is transformed gives us a complete picture of how \( T \) operates, enabling us to determine the transformation of any vector by decomposing it into these basis components.

When you have a vector \( \begin{bmatrix} a \ b \end{bmatrix} \) in \( \mathbb{R}^2 \), it can be written as a linear combination of the basis vectors:

• \( a \cdot \begin{bmatrix} 1 \ 0 \end{bmatrix} \)
• \( b \cdot \begin{bmatrix} 0 \ 1 \end{bmatrix} \)

By applying the linear transformation to these components, we find how \( T \) affects the entire vector.

A vector transformation is an operation that takes a vector as input and returns another vector as output. In the context of linear transformations, these operations follow specific rules and are defined by the transformation's effect on basis vectors.

In our example, the linear transformation \( T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} \) takes a 2D vector and transforms it into a 3D vector. This transformation is characterized by its rules on the standard basis vectors:

• \( T \begin{bmatrix} 1 \ 0 \end{bmatrix} = \begin{bmatrix} 1 \ 2 \ -1 \end{bmatrix} \)
• \( T \begin{bmatrix} 0 \ 1 \end{bmatrix} = \begin{bmatrix} 3 \ 0 \ 4 \end{bmatrix} \)

These rules help us compute the transformation of any vector by understanding how each component part undergoes the transformation. Any vector \( \begin{bmatrix} a \ b \end{bmatrix} \) can then be transformed by applying these rules and adding the resulting vectors.

Linear transformations can be conveniently represented as matrices. This representation makes computations simpler and more straightforward. For a linear transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \), the matrix will have dimensions matching the transition between these spaces, specifically a 3x2 matrix.

To derive the matrix of a transformation \( T \), we examine its effect on the basis vectors. Using the given transformations in our problem, our matrix \( A \) can be constructed by taking columns from:

• First column: \( T \begin{bmatrix} 1 \ 0 \end{bmatrix} = \begin{bmatrix} 1 \ 2 \ -1 \end{bmatrix} \)
• Second column: \( T \begin{bmatrix} 0 \ 1 \end{bmatrix} = \begin{bmatrix} 3 \ 0 \ 4 \end{bmatrix} \)

This results in matrix \( A = \begin{bmatrix} 1 & 3 \ 2 & 0 \ -1 & 4 \end{bmatrix} \).

Thus, any vector transformation \( T \begin{bmatrix} a \ b \end{bmatrix} \) can be easily calculated using matrix multiplication: \( A \begin{bmatrix} a \ b \end{bmatrix} \).

The hallmark of a linear transformation is its linearity property, which allows transformations to be simplified and calculated systematically. There are two main properties of linearity:

• **Additivity**: \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
• **Scalar Multiplication**: \( T(c\mathbf{u}) = cT(\mathbf{u}) \) for any scalar \( c \)

These properties mean that for vectors in \( \mathbb{R}^2 \), any vector \( \begin{bmatrix} a \ b \end{bmatrix} \) can be decomposed and individually transformed by its components.

In our exercise, to transform \( \begin{bmatrix} 5 \ 2 \end{bmatrix} \), we first handle each component separately, using known transformations. By linearity, combining these results through addition yields \( T\begin{bmatrix} 5 \ 2 \end{bmatrix} = 5T\begin{bmatrix} 1 \ 0 \end{bmatrix} + 2T\begin{bmatrix} 0 \ 1 \end{bmatrix} \), arriving at \( \begin{bmatrix} 11 \ 10 \ 3 \end{bmatrix} \).

This principle also applies to general vectors, allowing us to efficiently determine transformations without separately analyzing each individual vector case by case.