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Aleoa
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Is it possible to understand intuitively (without using a formal proof ) why a reflection is a linear function ?
Yes, @nuuskur's mirror is a good analogy: whether you stretch something or add lines with whatever angle in between, the mirror image does the same. The same can be said about rotations and stretches by their own. E.g. translations are not linear: if you stretch a line from one given and fixed point in a certain direction, a translated line segment will become something else than the not translated line segment. The point here is the fixed point.Aleoa said:Is it possible to understand intuitively (without using a formal proof ) why a reflection is a linear function ?
An intuitive linear transformation is a mathematical concept that describes a relationship between two sets of data that can be represented as straight lines. It involves transforming one set of data points into another set of data points using a linear function.
Intuitive linear transformation is unique because it preserves the linear relationship between the data sets. This means that the data points remain on the same line after the transformation is applied. Other types of transformations, such as non-linear transformations, do not necessarily preserve this relationship.
Intuitive linear transformation has many practical applications, including in economics, engineering, and physics. For example, it can be used to analyze the relationship between variables in an economic model, to design efficient transportation routes in engineering, and to describe the motion of objects in physics.
Intuitive linear transformation can be represented using matrix multiplication. The transformation matrix contains the coefficients of the linear function used to transform the data, and the input data points are represented as a vector. The resulting vector after multiplication represents the transformed data points.
Intuitive linear transformation is limited to linear relationships between data sets. This means that it may not accurately represent more complex relationships that are non-linear in nature. Additionally, it assumes that the data points are evenly spaced along the line, which may not always be the case in real-world data sets.