How to Find a Basis for the Image of a Linear Transformation

Linear algebra is a fundamental branch of mathematics that deals with linear equations and linear mappings between vector spaces. One of the core concepts in linear algebra is the image of a linear transformation, which refers to the set of all possible outputs produced by the transformation. In this article, we will explore a systematic method for finding a basis for the image of a linear transformation, with a focus on the mathematical steps and applications. This guide is particularly valuable for students and professionals working in mathematics, computer science, and related fields.

Steps to Find a Basis for the Image of a Linear Transformation

The process of determining a basis for the image of a linear transformation involves several key steps. Let's break down these steps and provide a detailed explanation for clarity.

1. Identify the Linear Transformation

The first step is to clearly define the linear transformation. For a given linear transformation $T: V to W$, where $V$ and $W$ are vector spaces, we need to express $T$ in terms of its action on a basis of $V$. If the transformation is represented by a matrix $A$ with respect to some bases, this matrix will be useful in subsequent steps.

2. Compute the Image

The second step is to determine the image of the linear transformation. The image, denoted as $text{Im}T$ or $TV$, is the set of all vectors $Tv$ for $v in V$. If the transformation is represented by a matrix $A$, the image can be found by examining the column space of $A$.

3. Form the Matrix of Images

In this step, we apply the transformation $T$ to each basis vector of $V$. Suppose ${e_1, e_2, ldots, e_n}$ is a basis for $V$. Then, we compute $T(e_1), T(e_2), ldots, T(e_n)$ and form a matrix $B$ whose columns are these images.

4. Row Reduce the Matrix

The next step involves performing row reduction (Gaussian elimination) on the matrix $B$ to bring it to its row echelon form (REF) or reduced row echelon form (RREF).

5. Identify Pivot Columns

In the row-reduced matrix, the columns corresponding to the pivot positions are linearly independent. These columns indicate the linearly independent vectors in the image of the transformation.

6. Extract the Basis

The final step is to extract the basis for the image of $T$. The vectors in the original matrix $B$ corresponding to the pivot columns in the row-reduced form form a basis for $text{Im}T$.

Example

Let us illustrate the method with an example using a linear transformation represented by the matrix:

$A begin{pmatrix} 1 2 3 4 5 6 end{pmatrix}$

Compute the Image

The columns of $A$ are $begin{pmatrix} 1 3 5 end{pmatrix}$ and $begin{pmatrix} 2 4 6 end{pmatrix}$.

Form the Matrix

Here, $B A$.

Row Reduce the Matrix

Row reducing $B$ gives:

$begin{pmatrix} 1 2 0 0 -2 0 0 0 0 end{pmatrix}$

Identify Pivot Columns

The pivot columns are the first and second columns.

Extract the Basis

The original vectors $begin{pmatrix} 1 3 5 end{pmatrix}$ and $begin{pmatrix} 2 4 6 end{pmatrix}$ form a basis for the image of $T$.

Conclusion

This method provides a structured approach for identifying a basis for the image of a linear transformation. By leveraging matrix representation and row reduction techniques, you can systematically find the basis for any linear transformation. This guide not only simplifies the process but also reinforces the understanding of key concepts in linear algebra, such as linear transformations, matrix representations, and the notion of image in vector spaces.