How to Find the Image of a Linear Transformation

Understanding and accurately identifying the image of a linear transformation is a fundamental skill in linear algebra. This article will guide you through the process with clear explanations, practical examples, and step-by-step instructions. By the end, you'll be equipped to find the image of any given linear transformation.

Understanding the Linear Transformation

A linear transformation (T: mathbb{R}^n rightarrow mathbb{R}^m) can be represented by a matrix (A) such that (Tmathbf{x} Amathbf{x}) for any vector (mathbf{x} in mathbb{R}^n). This matrix representation allows us to use matrix operations to determine the image of the transformation more efficiently.

Identifying the Matrix Representation

The first step is to identify the matrix (A) that represents the linear transformation. Consider the matrix (A) given by:

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

This matrix (A) will be used for our example throughout this article.

Find the Column Space

The image of the linear transformation is equivalent to the column space of the matrix (A). The column space is the span of the columns of (A). To proceed, you need to perform row reduction (Gaussian elimination) on (A) to bring it into its row echelon form or reduced row echelon form (RREF).

Determine the Basis for the Image

By identifying the pivot columns in the row-reduced matrix, you can determine a basis for the image of the transformation. Here’s how you do it:

Row Reduction: Perform row reduction on the matrix (A). Identify Pivot Columns: The columns in the original matrix that correspond to the pivot columns in the row-reduced matrix form a basis for the image of the transformation.

Express the Image

The image of the linear transformation can be expressed as:

(text{Im}T { Tmathbf{x} mid mathbf{x} in mathbb{R}^n } text{span} (text{columns of } A))

Example

Consider the linear transformation (T: mathbb{R}^3 rightarrow mathbb{R}^2) represented by the matrix:

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

Step 1: Identify the matrix (A). Step 2: Perform row reduction on (A):

(begin{pmatrix} 1 2 3 4 5 6 end{pmatrix} rightarrow begin{pmatrix} 1 2 3 0 -3 -6 end{pmatrix} rightarrow begin{pmatrix} 1 2 3 0 1 2 end{pmatrix} rightarrow begin{pmatrix} 1 0 -1 0 1 2 end{pmatrix})

The pivot columns are the first and second columns, indicating that the first and second columns of the original matrix (A) provide a basis for the image. Step 3: The image is then expressed as:

(text{Im}T text{span} left{ begin{pmatrix} 1 4 end{pmatrix}, begin{pmatrix} 2 5 end{pmatrix} right})

Conclusion

By following the steps outlined above, you can find the image of any linear transformation. Understanding and practicing this process will help you master the fundamental concepts of linear algebra and be well-prepared for more advanced topics.

Keywords

Linear transformation Image Matrix representation