A non-singular linear transformation is a mathematical operation that maps points in one vector space to corresponding points in another vector space, while preserving the linear structure of the original space. It is called non-singular because it does not result in any "singular" or degenerate transformations.
2.A singular linear transformation is one that results in a degenerate transformation, meaning it maps multiple points in the original space to the same point in the new space. This can result in a loss of information and is not invertible. On the other hand, a non-singular linear transformation preserves the structure and uniqueness of the original space and is invertible.
3.A linear transformation is non-singular if its determinant is non-zero. The determinant is a value that represents the scaling factor of the transformation and is calculated using the coefficients of the transformation matrix. If the determinant is zero, the transformation is singular and if it is non-zero, the transformation is non-singular.
4.Yes, a non-singular linear transformation can be represented by a matrix. The transformation matrix is a square matrix with a non-zero determinant and is used to perform the transformation on a vector. This matrix can also be inverted to perform the reverse transformation, making it a useful tool in linear algebra.
5.Non-singular linear transformations have various applications in fields such as computer graphics, image processing, and data compression. They are used to rotate, scale, and transform images and 3D objects in computer graphics. In data compression, non-singular transformations are used to reduce the size of data while preserving important information. They are also used in machine learning algorithms, such as principal component analysis, to reduce the dimensionality of data and extract relevant features.