Question

In Exercises 6-11, a linear transformation \(T\) is given. Find \([T]\). $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{l} x_{1}+x_{2} \\ x_{1}-x_{2} \end{array}\right] $$

Short answer

The matrix \([T]\) is \(\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\).

Step-by-step solution

Understand the Transformation

The linear transformation \(T\) takes a vector \(\begin{bmatrix} x_{1} \ x_{2} \end{bmatrix} \) and maps it to \(\begin{bmatrix} x_{1} + x_{2} \ x_{1} - x_{2} \end{bmatrix} \). Our task is to express this transformation as a matrix \([T]\).

Express Transformation in Terms of Basis Vectors

To find the matrix \([T]\), we express \(T\) in terms of standard basis vectors \(e_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\) and \(e_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix}\).

Apply Transformation to Basis Vectors

Calculate \(T(e_1)\) and \(T(e_2)\): \(T(e_1) = T\left(\begin{bmatrix} 1 \ 0 \end{bmatrix}\right) = \begin{bmatrix} 1 + 0 \ 1 - 0 \end{bmatrix} = \begin{bmatrix} 1 \ 1 \end{bmatrix}\)\(T(e_2) = T\left(\begin{bmatrix} 0 \ 1 \end{bmatrix}\right) = \begin{bmatrix} 0 + 1 \ 0 - 1 \end{bmatrix} = \begin{bmatrix} 1 \ -1 \end{bmatrix}\)

Formulate the Transformation Matrix

The matrix \([T]\) is formed by using the images of the basis vectors as its columns. Therefore, \([T] = \begin{bmatrix} T(e_1) & T(e_2) \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix}\).

Verify the Transformation Matrix

To ensure the correctness of \([T]\), multiply it by a general vector \(\begin{bmatrix} x_1 \ x_2 \end{bmatrix}\): \([T]\begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix}\begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} x_1 + x_2 \ x_1 - x_2 \end{bmatrix}\).The result matches \(T\)'s output, verifying our transformation matrix.

Key concepts

Basis Vectors

Basis vectors are the fundamental building blocks of vector spaces. In a simple two-dimensional space like the one we're considering, we typically use the standard basis vectors. These are usually defined as:

• \(e_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\)
• \(e_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix}\)

These vectors are called the basis vectors because any vector in this space can be expressed as a linear combination of them. For example, the vector \(\begin{bmatrix} x_1 \ x_2 \end{bmatrix}\) can be represented as \(x_1 e_1 + x_2 e_2\). The basis vectors essentially give us a coordinate system, which helps in expressing more complex transformations, such as the one given in the exercise.

By transforming these basis vectors using the linear transformation \(T\), we directly construct the columns of the transformation matrix \([T]\). This step is crucial in the solution of many linear algebra problems.

Transformation Matrix

The transformation matrix is a powerful tool that encapsulates how a linear transformation acts on a vector space. To find this matrix, we observe how the transformation \(T\) affects the basis vectors of the space.

• Calculate \(T(e_1)\), resulting in \(\begin{bmatrix} 1 \ 1 \end{bmatrix}\).
• Calculate \(T(e_2)\), resulting in \(\begin{bmatrix} 1 \ -1 \end{bmatrix}\).

The transformation matrix \([T]\) is then composed of these transformed basis vectors. They become the columns of the matrix:

• \([T] = \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix}\)

This matrix allows us to perform the transformation \(T\) on any vector in the space simply by using matrix multiplication. It is a compact representation of the transformation operation that succinctly communicates how input vectors are altered.

Matrix Multiplication

Matrix multiplication is an algebraic operation that applies transformations to vectors. Given a transformation matrix \([T]\) and a vector \(\begin{bmatrix} x_1 \ x_2 \end{bmatrix}\), the matrix multiplication \[[T]\begin{bmatrix} x_1 \ x_2 \end{bmatrix}\]executes the linear transformation on the vector.

• For \([T] = \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix}\), multiplying with \(\begin{bmatrix} x_1 \ x_2 \end{bmatrix}\) gives: \(\begin{bmatrix} 1 \times x_1 + 1 \times x_2 \ 1 \times x_1 - 1 \times x_2 \end{bmatrix}\).

This operation results in the vector \(\begin{bmatrix} x_1 + x_2 \ x_1 - x_2 \end{bmatrix}\), confirming that the transformation matrix captures the intended action of \(T\).

Matrix multiplication leverages the linearity of transformations, allowing a consistent application of geometric operations across all vectors in the space. Understanding how matrix multiplication operates is key in solving many linear transformations and grasping their effects.