91Ó°ÊÓ

A matrix representation is a powerful way to encapsulate what a linear transformation does. In this exercise, the linear transformation \(T\) maps vectors from \(\mathbb{R}^2\) to \(\mathbb{R}^3\). To find the matrix that represents \(T\), start by applying \(T\) to each vector in the standard basis of the domain (\(\mathbb{R}^2\)).

Each transformation result is then expressed as a linear combination of the standard basis in the codomain (\(\mathbb{R}^3\)). The coefficients of these linear combinations form the columns of the matrix.

• Apply \(T\) to the first basis vector, \((1, 0)\), resulting in \((1, 0, 2)\).
• Apply \(T\) to the second basis vector, \((0, 1)\), yielding \((1, 0, -1)\).

Thus, the matrix representation of \(T\) is \(\begin{pmatrix}1 & 1 \ 0 & 0 \ 2 & -1\end{pmatrix}\). This matrix transforms any vector in \(\mathbb{R}^2\) to \(\mathbb{R}^3\) by multiplying it, thus performing the linear transformation.

The standard basis is foundational in linear algebra, providing a default way to describe any vector space. For a vector space, its standard basis contains vectors like \(\vec{e_1}, \vec{e_2}, \ldots\), where each unit vector has a single component equal to one and all others zero. In our exercise, we're dealing with different spaces.

• The standard basis for \(\mathbb{R}^2\) includes \(\{(1, 0), (0, 1)\}\).
• For \(\mathbb{R}^3\), it's \(\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}\).

The linear transformation \(T\) maps each vector in \(\mathbb{R}^2\) to \(\mathbb{R}^3\), using these standard basis vectors. Understanding the transformation of basis vectors assists in constructing the matrix representation of \(T\). In this case, the application of \(T\) on the standard basis of \(\mathbb{R}^2\) helped directly formulate our transformation matrix.

Domain and codomain are crucial concepts in understanding functions, including linear transformations. A function's domain is the set of 'input' values, while the codomain is the set of potential 'output' values. For the linear transformation \(T\), specified in this exercise:

• The domain is \(\mathbb{R}^2\), meaning all vectors made up of two real numbers.
• The codomain is \(\mathbb{R}^3\), covering vectors composed of three real numbers.

These two roles determine how the function maps vectors between spaces. The domain and codomain not only define the nature of the input and output for \(T\) but also help identify the transformation's dimensions. Understanding these spaces is vital when analyzing or using a transformation, especially in more advanced applications where the transformation properties, such as rank or nullity, are explored.