Define linear transformation in complex analysis
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In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mappingV → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
An important special case is when V = W, in which case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not.
A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension); for instance it maps a plane through the origin to a plane, straight line or point. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.
In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory it is a morphismin the category of modules over a given ring
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An important special case is when V = W, in which case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not.
A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension); for instance it maps a plane through the origin to a plane, straight line or point. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.
In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory it is a morphismin the category of modules over a given ring
Hope this answer helps to you☺️☺️
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A function of the form f(z)=az+b where a and b are complex constants is called a linear transformation. show that every linear transformation can be expressed as the composition of a magnification, a ratation, and a translation. deduce from this that a linear transformation maps lines to lines and circles to circles.
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