Question

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

Short answer

The matrix representation of \( T \) is \( \begin{bmatrix} 1 & 0 & 3 \\ 1 & 0 & -1 \\ 1 & 0 & 1 \end{bmatrix} \).

Step-by-step solution

Understand the Transformation

We are given a linear transformation \( T \) that maps a vector \( \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \) to a new vector \( \begin{bmatrix} x_1 + 3x_3 \ x_1 - x_3 \ x_1 + x_3 \end{bmatrix} \). The goal is to find the matrix representation \([T]\) of this transformation.

Apply Transformation to Standard Basis Vectors

To find \([T]\), apply the transformation \(T\) to each of the standard basis vectors \( e_1, e_2, e_3 \) of \( \mathbb{R}^3 \). These vectors are: - \( e_1 = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix} \) - \( e_2 = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix} \) - \( e_3 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \).

Compute \(T(e_1)\)

Substitute \( e_1 = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix} \) into the transformation equation: \[ T\left( \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix} \right) = \begin{bmatrix} 1+3(0) \ 1-0 \ 1+0 \end{bmatrix} = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}. \] This vector is the first column of \([T]\).

Compute \(T(e_2)\)

Substitute \( e_2 = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix} \) into the transformation equation: \[ T\left( \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix} \right) = \begin{bmatrix} 0+3(0) \ 0-0 \ 0+0 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}. \] This vector is the second column of \([T]\).

Compute \(T(e_3)\)

Substitute \( e_3 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \) into the transformation equation: \[ T\left( \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \right) = \begin{bmatrix} 0+3(1) \ 0-1 \ 0+1 \end{bmatrix} = \begin{bmatrix} 3 \ -1 \ 1 \end{bmatrix}. \] This vector is the third column of \([T]\).

Construct the Matrix \([T]\)

Combine the results from the previous steps to form the matrix representation of \(T\). The matrix \([T]\) is formed by taking the vectors \( \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix} \), \( \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix} \), and \( \begin{bmatrix} 3 \ -1 \ 1 \end{bmatrix} \) as columns:\[ [T] = \begin{bmatrix} 1 & 0 & 3 \ 1 & 0 & -1 \ 1 & 0 & 1 \end{bmatrix}. \]

Key concepts

Matrix Representation

A matrix representation is a powerful way to visualize and compute the effects of a linear transformation. When you have a linear transformation like the one described by the function \( T \), which takes a vector in \( \mathbb{R}^3 \) and maps it to another vector, creating its matrix representation \( [T] \) involves expressing this transformation in the form of a matrix. This matrix will allow you to easily apply the transformation to any vector in \( \mathbb{R}^3 \) by simple matrix multiplication.

To find this matrix, we apply the transformation to each of the standard basis vectors, which are fundamental building blocks in vector spaces. Each result becomes a column in the transformation matrix, giving us a complete picture of how \( T \) operates. This method makes it straightforward to predict the output of \( T \) for any input vector.

Standard Basis Vectors

Standard basis vectors are the building blocks of vector spaces. In the context of \( \mathbb{R}^3 \), they are the vectors:

• \( e_1 = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix} \)
• \( e_2 = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix} \)
• \( e_3 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \)

These vectors serve as a convenient and familiar starting point for defining any other vector in \( \mathbb{R}^3 \) because any vector can be expressed as a unique linear combination of \( e_1, e_2, \) and \( e_3 \).

Using the standard basis vectors simplifies the process of finding a transformation matrix. By seeing how the transformation \( T \) transforms each of these basis vectors, you can understand how it will transform any vector. This method reduces the problem to working with simple, individual vectors instead of more complex ones.

Vector Spaces

Vector spaces are fundamental mathematical structures used in linear algebra and beyond. They consist of a set of vectors, where you can perform vector addition and scalar multiplication. \( \mathbb{R}^3 \) is an example of a vector space, representing all 3-dimensional vectors.Vector spaces have a set of defining properties:

• Closure under addition.
• Closure under scalar multiplication.
• Existence of a zero vector.
• Existence of additive inverses.
• Associativity and commutativity of addition.
• Distributive property of scalar multiplication over vector addition.

Understanding vector spaces is crucial when working with linear transformations, as they provide the framework within which these transformations operate. Each vector in the space can be transformed, and the results still belong to the same or another specific vector space, preserving the underlying structure.

Transformation Matrices

Transformation matrices are the matrices that represent linear transformations like \( T \). Once created, these matrices allow for straightforward computation of output vectors by simply multiplying the transformation matrix by input vectors.

For a transformation matrix \( [T] \), each column represents the image of the standard basis vectors under the transformation. This method ensures that any vector transformed by \( [T] \) will result in the correct mapping according to \( T \). This relationship transforms what could be a complex computation into a feasible set of operations using matrix algebra.

Transformation matrices are incredibly useful not just in pure mathematics, but in practical applications such as computer graphics, data science, and engineering, where they facilitate fast and efficient computations for large sets of data.