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How do you find the range of a transformation function?

Published in Linear Algebra 3 mins read

Finding the range of a transformation function involves determining all possible output vectors that can be produced by the transformation. This often requires understanding the function's properties and how it affects the input vectors.

General Approach:

  1. Understand the Transformation: Clearly define the transformation function. This includes knowing its input and output vector spaces (e.g., transforms vectors from R4 to R3).

  2. Consider the Input Space: Understand the constraints or characteristics of the input vectors. Are there any limitations on the possible values of the input?

  3. Determine the Image of Basis Vectors (for Linear Transformations): If the transformation is linear, you can find the image (output) of a basis for the input space. These images will span the range.

  4. Express the Range as a Span: The range is the span of the images of the basis vectors. This means any linear combination of these image vectors is in the range.

  5. Find a Basis for the Range: The image vectors found in step 4 might be linearly dependent. If so, find a linearly independent subset that still spans the same space. This forms a basis for the range.

  6. Determine the Dimension of the Range: The number of vectors in the basis for the range is the dimension of the range, also known as the rank of the transformation.

  7. Describe the Range: Express the range as the set of all linear combinations of the basis vectors, or in terms of equations that must be satisfied by the components of the output vectors.

Example (Linear Transformation):

Suppose we have a linear transformation T: R4 -> R3. To find its range:

  1. Let's say T(v) = Av, where A is a 3x4 matrix representing the transformation.

  2. Find the image of the standard basis vectors e1, e2, e3, and e4 of R4. These images are just the columns of the matrix A.

  3. Range(T) = Span{T(e1), T(e2), T(e3), T(e4)} The range is the span of the columns of A (also known as the column space of A).

  4. Reduce the Columns of A to find a basis: Use Gaussian elimination or other methods to find a linearly independent set of columns in A. These columns form a basis for the range.

Non-Linear Transformations:

Finding the range of a non-linear transformation can be more difficult. There isn't a single, universally applicable method. You might need to:

  • Analyze the function's behavior directly.
  • Consider the level sets of the function.
  • Use calculus to find critical points and extreme values.
  • Rely on specific knowledge of the function's properties.

In Summary:

Finding the range of a transformation function generally involves determining all possible output vectors. For linear transformations, this often means finding the span of the images of a basis for the input space (the column space of the matrix). For non-linear transformations, more specialized techniques might be necessary.

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