W be a linear transformation, where V and W are vector spaces. The linear transformation T maps vectors from V to W.

In more detail, for every vector v in V, there is a corresponding vector T(v) in W. T preserves the linear structure, which means that for any vectors v and u in V and any scalar c, T(c*v + u) = c*T(v) + T(u).

Linear transformations are important in linear algebra and have many applications in various fields of mathematics, physics, and engineering. They are used to represent and analyze various types of transformations, such as rotations, translations, and scaling.

Some common examples of linear transformations include:

1. Rotation: A rotation of a vector in 2D or 3D space is a linear transformation. It preserves the length and angle between vectors.

2. Projection: A projection of a vector onto another vector or subspace is a linear transformation. It projects a vector onto a lower-dimensional subspace.

3. Scaling: A scaling transformation multiplies each component of a vector by a scalar. It can stretch or shrink the vector in different directions.

4. Translation: A translation transformation moves a vector by a fixed amount in a specific direction. It adds a constant vector to the original vector.

Linear transformations can be represented by matrices. The matrix representation of a linear transformation depends on the choice of bases for the vector spaces V and W. The matrix provides a convenient way to compute the image of a vector under the transformation.

Overall, linear transformations play a fundamental role in linear algebra and provide a powerful tool for analyzing and solving problems involving vectors and vector spaces.