41. While studying compositions of linear transformations , Arthur Cayley was led to define matrix multiplication and inverses. 42. Linear transformations are global in nature, thus, they cannot model local geometric differences between images.43. The reflection of a point " x " about this hyperplane is the linear transformation : 44. Mahalanobis distance is preserved under full-rank linear transformations of the space spanned by the data. 45. Accordingly, every invertible linear transformation of a finite-dimensional linear topological space is a homeomorphism. 46. Galilean scale relativity involves linear transformations a constant fractal dimension, self-similarity and scale invariance. 47. The theory of ordinary similarity arises as a result of studying linear transformations referred to different bases. 48. It is not classified as a linear transformation , but is instead known as an affine transformation. 49. All of the integral transforms can be expressed as linear transformations ; that is, as matrices. 50. However, often the definition of the G�teaux differential also requires that it be a continuous linear transformation .
