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coco richie
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Every definition I find on Google utilizes vector spaces and mapping which are mathematical. If you have to use vector spaces and mapping please explain the mathematics behind it. Thanks.
In fact, if f and g are linear, then f(g(x)) = g(f(x)), so we may as well write fg(x) = gf(x).
slider142 said:These two properties define a linear transformation, no matter how abstract the field of study is.
coco richie said:Just to be clear slider, are you referring to multiplication and addition as the two properties of the functions that define linear transformation? What other choices do we have that makes a function "non-linear"?
Studiot said:
coco richie said:How would you word this equation? Is it f times g(x) is equal to g times f(x)?
coco richie said:Every definition I find on Google utilizes vector spaces and mapping which are mathematical. If you have to use vector spaces and mapping please explain the mathematics behind it. Thanks.
A translation is *not* a linear transformation; it is an affine transformation.
slider142 said:In fact, if f and g are linear, then f(g(x)) = g(f(x)), so we may as well write fg(x) = gf(x).
Rasalhague said:Possible interpretation: composition of linear functions is commutative. Counter example: rotations in R3. So I guess it's not that... Could you disambiguate?
A linear transformation is a mathematical operation that takes in a set of numbers or vectors and produces a new set of numbers or vectors by following specific rules. It is commonly used in fields such as physics, engineering, and data analysis to manipulate data in a systematic way.
There are two main characteristics of a linear transformation: it preserves addition and scalar multiplication. This means that the result of adding two numbers or vectors and then applying the linear transformation is the same as applying the linear transformation to each number or vector separately and then adding them together. Similarly, multiplying a number or vector by a constant and then applying the linear transformation is the same as applying the linear transformation and then multiplying the result by the constant.
A linear transformation can be represented in different ways, depending on the context. In general, it is represented as a matrix, which is a rectangular array of numbers. The size of the matrix is determined by the number of inputs and outputs of the transformation. For example, a 2x2 matrix represents a linear transformation that takes in 2-dimensional input and produces 2-dimensional output.
The main difference between a linear and non-linear transformation is that a linear transformation follows the rules of linearity, while a non-linear transformation does not. This means that the result of applying a linear transformation to a combination of inputs can be obtained by applying the linear transformation to each input separately and then combining the results. In contrast, a non-linear transformation does not follow this rule and can produce unexpected or complex results.
Linear transformations have various applications in real life, including image and signal processing, machine learning, and economics. For example, in image processing, linear transformations are used to resize, rotate, or enhance images. In economics, linear transformations are used to model relationships between variables and make predictions. In machine learning, linear transformations are used to transform and analyze data to make decisions or predictions.