Question

A linear transformation \(T\) is given. Find \([T]\). $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\right)=\left[\begin{array}{c} x_{1}+2 x_{2}-3 x_{3} \\ 0 \\ x_{1}+4 x_{3} \\ 5 x_{2}+x_{3} \end{array}\right] $$

Short answer

The transformation matrix \([T]\) is \( \begin{bmatrix} 1 & 2 & -3 \\ 0 & 0 & 0 \\ 1 & 0 & 4 \\ 0 & 5 & 1 \end{bmatrix} \).

Step-by-step solution

Define the Transformation Matrix

To find the matrix representation of the linear transformation \( T \), we denote \( [T] \) as the transformation matrix. Our goal is to express \( T \) in terms of standard basis vectors \( e_1, e_2, e_3 \) and find \([T] \) such that \( [T] \cdot [x_1, x_2, x_3]^T \) produces the result of \( T \).

Apply T to Standard Basis Vectors

Calculate \( T(e_1), T(e_2), T(e_3) \), where \( e_1 = [1,0,0]^T \), \( e_2 = [0,1,0]^T \), \( e_3 = [0,0,1]^T \). We compute:\[ T(e_1) = \begin{bmatrix} 1 \ 0 \ 1 \ 0 \end{bmatrix}, \quad T(e_2) = \begin{bmatrix} 2 \ 0 \ 0 \ 5 \end{bmatrix}, \quad T(e_3) = \begin{bmatrix} -3 \ 0 \ 4 \ 1 \end{bmatrix} \]

Formulate the Transformation Matrix

The columns of the matrix \([T]\) are the transformations applied to the standard basis vectors. Therefore, we construct \([T]\) as follows:\[ [T] = \begin{bmatrix} 1 & 2 & -3 \ 0 & 0 & 0 \ 1 & 0 & 4 \ 0 & 5 & 1 \end{bmatrix} \]

Verify the Transformation Matrix

Check the transformation matrix by verifying that applying \([T]\) to \([x_1, x_2, x_3]^T\) results in the transformed vector described by \( T\). The multiplication gives:\[ [T] \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 1 & 2 & -3 \ 0 & 0 & 0 \ 1 & 0 & 4 \ 0 & 5 & 1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} x_1+2x_2-3x_3 \ 0 \ x_1+4x_3 \ 5x_2+x_3 \end{bmatrix} \] which matches exactly with the transformation \( T \).

Key concepts

Matrix Representation

When dealing with linear transformations like the one presented in the exercise, a matrix is often used to represent this transformation concisely. This matrix, often denoted as [T], encapsulates how each input vector is transformed in terms of the standard basis vectors.

Understanding the matrix representation of a transformation is crucial because it provides a straightforward way to apply the transformation to any vector in the domain. The process involves matrix-vector multiplication, which is computationally efficient and easy to understand. Once the transformation matrix is found, it can be reused to transform any vector from the input space to the output space.

In summary, matrix representation is essential for simplifying the process of applying a transformation to different vectors, making it a fundamental concept in linear algebra.

Standard Basis Vectors

Standard basis vectors are the building blocks for representing vectors in any vector space. In three-dimensional space, these are usually defined as \(e_1 = [1, 0, 0]^T\), \(e_2 = [0, 1, 0]^T\), and \(e_3 = [0, 0, 1]^T\). Each of these vectors has a "1" in one position and "0" in the others, making it very useful for analyzing and constructing other vectors.

In the context of transformations, standard basis vectors allow us to see how a transformation affects the space's axes directly. When you apply a linear transformation to these basis vectors, the resulting vectors form the columns of the transformation matrix. This shows how each basis vector is affected individually by the transformation.

Therefore, standard basis vectors not only represent the basic directions in a vector space but also help to uncover the core effects of a transformation through its matrix.

Transformation Matrix

The transformation matrix is the key outcome of applying a linear transformation to standard basis vectors. Once you have computed how each basis vector transforms, each result forms a column of this matrix.

In this exercise, applying the given linear transformation to \(e_1, e_2,\) and \(e_3\), we obtained respective vectors:

• \(T(e_1) = \begin{bmatrix} 1 & 0 & 1 & 0 \end{bmatrix}\)
• \(T(e_2) = \begin{bmatrix} 2 & 0 & 0 & 5 \end{bmatrix}\)
• \(T(e_3) = \begin{bmatrix} -3 & 0 & 4 & 1 \end{bmatrix}\)

Once these vectors are aligned as columns, they form the transformation matrix \([T]\), which is: \(\begin{bmatrix} 1 & 2 & -3 \ 0 & 0 & 0 \ 1 & 0 & 4 \ 0 & 5 & 1 \end{bmatrix}\).

This transformation matrix now represents the effect of \(T\)on all possible input vectors when applied through matrix multiplication, encapsulating the transformation efficiently and completely.

Verification of Transformation Matrix

Verifying the correctness of the transformation matrix is a critical step to ensure accuracy. Once the transformation matrix is derived, you can check it by applying it to any vector and see if it matches the transformation defined originally.

In this exercise, the transformation matrix \([T]\) was verified by applying it to a generic vector \([x_1, x_2, x_3]^T\). Multiplying \([T]\) by this vector should give you the defined transformation output:\( \begin{bmatrix} x_1 + 2x_2 - 3x_3 \ 0 \ x_1 + 4x_3 \ 5x_2 + x_3 \end{bmatrix} \).

This process ensures that each component is correctly represented and consistent with the transformation's original definition. Verification not only confirms the process but also boosts confidence in using the transformation matrix for further calculations.