Range

Definition

The range of a linear transformation is the set of all possible output vectors that can be produced from applying the transformation to input vectors from its domain. It essentially reflects how the transformation can stretch, compress, or alter the input space and is a crucial concept in understanding the behavior of linear mappings. The range provides insights into the image of the transformation and its effectiveness in mapping the entire vector space.

5 Must Know Facts For Your Next Test

1. The range of a linear transformation is always a subspace of the codomain, meaning it satisfies properties like closure under addition and scalar multiplication.
2. To find the range, you can express the output vectors in terms of linear combinations of the transformation's basis vectors.
3. If the linear transformation is represented by a matrix, the range can be determined by identifying the column space of that matrix.
4. The dimension of the range is referred to as the rank of the transformation, which is an important concept in understanding its effectiveness.
5. A linear transformation is said to be onto (surjective) if its range equals its codomain, meaning every vector in the codomain can be expressed as an output.

Review Questions

• How can you determine if a linear transformation is onto based on its range?
  • A linear transformation is onto if its range matches the entire codomain. This means that for every vector in the codomain, there exists at least one vector in the domain that maps to it under the transformation. To check this, you can analyze the column space of the transformation's associated matrix and see if it spans the whole codomain.
• In what ways does understanding the range of a linear transformation help in solving systems of linear equations?
  • Understanding the range helps identify whether a system of equations has solutions and how many solutions there may be. If the range covers all possible outputs needed for a consistent system, then solutions exist. Additionally, knowing the dimension of the range gives insight into whether solutions are unique or if there are infinitely many, based on how it relates to other dimensions within the context of the equations.
• Evaluate how changes in input dimensions affect the range and rank of a linear transformation.
  • Changes in input dimensions can significantly impact both the range and rank of a linear transformation. When input dimensions increase, it may lead to a larger or potentially infinite range if additional independent directions are introduced. However, if some input dimensions are dependent, it could limit the rank since fewer unique output directions would be possible. Thus, analyzing these changes allows us to understand how transformations behave under varying conditions and their implications on vector spaces.

Related terms

Linear Transformation: A function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Image: The set of all output vectors produced by applying a linear transformation to the vectors in its domain.

Dimension: A measure of the size of a vector space, representing the number of vectors in a basis for that space.