Question

A linear transformation \(T\) is given. Find \([T]\). $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{c} x_{1}+2 x_{2} \\ 3 x_{1}-5 x_{2} \\ 2 x_{2} \end{array}\right] $$

Short answer

The transformation matrix \([T]\) is \( \begin{bmatrix} 1 & 2 \\ 3 & -5 \\ 0 & 2 \end{bmatrix} \).

Step-by-step solution

Understand the Transformation

The linear transformation \( T \) is defined for a 2-dimensional vector \( \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) resulting in a 3-dimensional output vector. It maps vectors from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \).

Find the Output for Basis Vectors

Calculate \( T \) for the standard basis vectors \( \begin{bmatrix} 1 \ 0 \end{bmatrix} \) and \( \begin{bmatrix} 0 \ 1 \end{bmatrix} \) of \( \mathbb{R}^2 \). This will give the columns of the transformation matrix \([T]\).

Calculate \( T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) \)

Apply the transformation rule: \( T \left( \begin{bmatrix} 1 \ 0 \end{bmatrix} \right) = \begin{bmatrix} 1 + 2 \times 0 \ 3 \times 1 - 5 \times 0 \ 2 \times 0 \end{bmatrix} = \begin{bmatrix} 1 \ 3 \ 0 \end{bmatrix} \).

Calculate \( T \left( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) \)

Apply the transformation rule: \( T \left( \begin{bmatrix} 0 \ 1 \end{bmatrix} \right) = \begin{bmatrix} 0 + 2 \times 1 \ 3 \times 0 - 5 \times 1 \ 2 \times 1 \end{bmatrix} = \begin{bmatrix} 2 \ -5 \ 2 \end{bmatrix} \).

Form the Transformation Matrix \([T]\)

Combine the results from Step 3 and Step 4 as columns to form the matrix: \[[T] = \begin{bmatrix} 1 & 2 \ 3 & -5 \ 0 & 2\end{bmatrix}\]

Key concepts

Transformation Matrix

In linear algebra, a transformation matrix is an essential computational tool. It is used to represent linear transformations between vector spaces. This matrix acts as a blueprint that maps input vectors to output vectors according to specified rules. For instance, consider the linear transformation \( T \) defined from a 2D vector \( \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) into a 3D vector space. After applying the transformation matrix, the 2D vectors are converted into 3D output vectors through linear mapping.
The transformation matrix \([T]\) is constructed by evaluating the function on the input space's standard basis vectors. This involves solving for \( T \left( \begin{bmatrix} 1 \ 0 \end{bmatrix} \right) \) and \( T \left( \begin{bmatrix} 0 \ 1 \end{bmatrix} \right)\). Each result becomes a column in the transformation matrix.

• The output from \( T \left( \begin{bmatrix} 1 \ 0 \end{bmatrix} \right) \) contributes the first column.
• The output from \( T \left( \begin{bmatrix} 0 \ 1 \end{bmatrix} \right) \) contributes the second column.

The final transformation matrix \([T]\), therefore, is the aggregation of these column vectors. It's crucial to understand how this matrix provides a way to perform transformations computationally, ensuring consistent output for any input vector.

Basis Vectors

Basis vectors form the foundation of a vector space. They provide a reference framework from which every other vector in the space can be described. In our exercise, the basis vectors for \( \mathbb{R}^2 \) are \( \begin{bmatrix} 1 \ 0 \end{bmatrix} \) and \( \begin{bmatrix} 0 \ 1 \end{bmatrix} \), known as the standard basis vectors.
Using these vectors allows us to easily evaluate linear transformations like \( T \). By applying \( T \) to the standard basis vectors, we derive the columns of the transformation matrix. Essentially, these basis vectors define directions within the vector space, contributing to an intuitive understanding of how transformations influence vectors.

• Standard basis vectors provide a simple and consistent way to construct transformation matrices.
• They simplify calculations, allowing focus on the transformational effect alone.

Hence, mastering basis vectors is vital for transforming and analyzing vectors within various dimensions. Always remember, they form the skeleton upon which vector spaces—and transformations—are built.

Vector Spaces

Vector spaces are a fundamental construct in mathematics that contain vectors. They are defined over a field, often real numbers, and provide a setting for vector addition and scalar multiplication. Understanding vector spaces is critical for grasping concepts like linear transformations.
In our exercise, we see a transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \), exemplifying how vectors can be transformed between spaces of different dimensions. A vector space must be equipped with a set of operations that satisfy certain axioms or rules. These include associative, commutative, and distributive properties.

• Vector spaces permit analysis in higher dimensions, enabling complex transformations.
• They set the stage for exploring linear mappings, function spaces, and more.

Comprehending vector spaces allows for deeper insights into linear algebra's capabilities and how transformations reshape vectors within or across these multi-dimensional landscapes.