verty said:Do you understand that this is true for any function ever? That is how functions are defined, the range is a subset of the codomain.
A David said:Any mapping ever, really.
A linear transformation is a function that maps one vector space to another in a way that preserves the structure of the original vector space. This means that the properties of vectors, such as addition and scalar multiplication, are maintained under the transformation.
Linear transformations can change the orientation, magnitude, and direction of vectors. They can also stretch, rotate, and reflect vectors depending on the specific transformation being applied.
A linear transformation follows the principles of linearity, which means that the output is a linear combination of the input. Nonlinear transformations, on the other hand, do not follow these principles and can produce curved or distorted output.
Linear transformations are used in a variety of fields, including physics, engineering, and computer graphics. They are used to analyze data, model physical systems, and create visual effects in movies and video games.
A linear transformation can be represented using a matrix, which is a rectangular array of numbers. The matrix contains the coefficients of the linear transformation and can be used to apply the transformation to vectors by multiplying them together.