LINEAR TRANSFORMATION Meaning and
Definition
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A linear transformation, also known as a linear map or linear function, refers to a mathematical operation that preserves the properties of linearity. It is a mapping between two vector spaces, usually denoted as V and W, that assigns each vector in V to a corresponding vector in W, while respecting the linear structure of both vector spaces.
More formally, a linear transformation T: V → W is a function that satisfies two fundamental properties: preservation of addition and preservation of scalar multiplication. Preservation of addition means that for any two vectors u and v in V, the transformation T(u + v) will be equal to T(u) + T(v) in W. Similarly, preservation of scalar multiplication states that for any vector u in V and any scalar c, T(cu) will be equal to cT(u) in W.
Linear transformations possess several important characteristics. They are said to be homogeneous, meaning that they scale with the input, and they also exhibit additivity, as they preserve vector addition. Furthermore, the transformation must preserve the zero vector, which ensures the linearity of both vector spaces.
Linear transformations find various applications in mathematics, physics, computer science, and engineering. They play a crucial role in fields such as linear algebra, functional analysis, and differential equations. By defining relationships between vector spaces and being able to describe transformations between them, linear transformations provide a powerful tool for solving problems and understanding the underlying structure of mathematical systems.
Common Misspellings for LINEAR TRANSFORMATION
- kinear transformation
- pinear transformation
- oinear transformation
- lunear transformation
- ljnear transformation
- lknear transformation
- lonear transformation
- l9near transformation
- l8near transformation
- libear transformation
- limear transformation
- lijear transformation
- lihear transformation
- linwar transformation
- linsar transformation
- lindar transformation
- linrar transformation
- lin4ar transformation
- lin3ar transformation
- linezr transformation
Etymology of LINEAR TRANSFORMATION
The word "linear" comes from the Latin word "linearis", which means "consisting of lines" or "belonging to lines". The term "transformation" comes from the Latin word "transformatio", which means "a changing or fashioning". In mathematics, a linear transformation refers to a mapping between two vector spaces that preserves linearity or straightness, and this concept dates back to ancient Greece. However, the explicit term "linear transformation" was coined in the early 20th century to describe this mathematical operation.
