How do you know if a transformation is linear?
Understand the Problem
The question is asking for the criteria or characteristics that determine whether a given transformation is linear or not. This typically involves understanding the properties of linear transformations, such as preserving addition and scalar multiplication.
Answer
Satisfy additivity (T(x + y) = T(x) + T(y)) and homogeneity (T(kx) = kT(x)).
To determine if a transformation T is linear, check if it satisfies both additivity (T(x + y) = T(x) + T(y)) and homogeneity (T(kx) = kT(x)) for all vectors x and y, and scalars k.
Answer for screen readers
To determine if a transformation T is linear, check if it satisfies both additivity (T(x + y) = T(x) + T(y)) and homogeneity (T(kx) = kT(x)) for all vectors x and y, and scalars k.
More Information
Linear transformations map straight lines to straight lines and preserve vector space operations.
Tips
A common mistake is to check only one of the conditions. Both additivity and homogeneity must be satisfied for the transformation to be linear.
Sources
- Fundamentals of Matrix Algebra - LibreTexts - math.libretexts.org
- Math Insight: Linear Transformations - mathinsight.org
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