From the course: Machine Learning Foundations: Linear Algebra
Transforming to the new basis
From the course: Machine Learning Foundations: Linear Algebra
Transforming to the new basis
- [Instructor] We have explored matrix transformation in a simple example. Let's look at the fundamentals and rules behind it. Matrix transformations are a special class of functions that arise from matrix multiplication. In mathematics, we define an ordered n-tuple as a sequence of n real numbers and a solution of a linear system in n unknowns that can be written as x1 = s1, x2 = s2 until xn = sn. It can be expressed as an ordered n-tuple, s1, s2, until sn. We do know the set of all ordered n-tuples of real numbers with a bold capital letter R and superscript n. The elements for Rn are called vectors. Standard basis vectors are denoted as e1, e2, until en. All other vectors in Rn can be written in exactly one way is a linear combination of basis vectors. So if you have a vector x, it can be written as x = x1e1 + x2e2 + xnen. As matrix transformation is a special class of function, we usually use the letter T to denote it. We can write matrix transformation from Rn to Rm as. If we think about matrix transformations that arise from linear systems, we can write them as y = Ax. So matrix transformation maps a vector x in Rn into the vector y in Rm by multiplying x with A. We can write it down as y = Ta of x. We have learned how to transform a vector b to any basis as long as we have the basis vectors of a new vector space. We can follow these three steps. Transform the vector b to our standard coordinate system using the appropriate transformation matrix A that results in b prime, Ab = b prime. Perform a custom transform on b prime. Let's say we have a transformation represented by the matrix R in the standard coordinate system giving us a rotated vector c prime, Rb prime = c prime. Transform c prime back to the alternate coordinate system using the inverse of A that will result in the vector c. Vector c is a transformation of the vector b in the alternative coordinate system.
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