A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the structure of the space. This means that the transformation maintains the properties of addition and scalar multiplication.
To determine if a function is a linear transformation, you can check if it satisfies two conditions: additivity and homogeneity. Additivity means that the function must preserve the property of vector addition, while homogeneity means that the function must preserve the property of scalar multiplication.
Some common examples of linear transformations include translations, rotations, reflections, and scaling. These transformations are commonly used in geometry and computer graphics.
Yes, a function can be a linear transformation in one vector space but not in another. This is because different vector spaces may have different properties that the function needs to preserve.
To prove that a function is a linear transformation, you can use the properties of additivity and homogeneity. You can also use mathematical techniques such as using matrices and solving systems of equations to show that the function preserves the properties of vector addition and scalar multiplication.