What is the purpose of a transformation matrix in applied linear algebra?

The purpose of a transformation matrix in applied linear algebra is to transform vectors to different vector spaces. A transformation matrix represents a linear transformation, which is a function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. This means that when you apply a transformation matrix to a vector, you are effectively changing its position or orientation in the vector space.

Transformation matrices can represent various operations, such as rotations, translations, scaling, and shearing. Each type of transformation corresponds to a specific matrix that when multiplied by a vector modifies it according to the nature of that transformation. For example, in 2D space, a rotation matrix can rotate a vector around the origin, while a scaling matrix can resize it.

The other options describe actions or concepts that do not directly relate to the primary agent of transformation in applied linear algebra. While adding matrices and vectors is a fundamental operation in linear algebra, it does not capture the essence of what a transformation matrix does. Similarly, eigenvalues and eigenvectors are concepts that arise from linear transformations but are not the purpose of a transformation matrix itself. Lastly, transformation matrices are not limited to performing only scalar multiplications, as they encompass a broader range of operations involving vectors.