From Variables to Matrices in Mathematics and Machine Learning
Transformation is a fundamental change in the nature of something. This may be as simple as an outward change, such as a physical transformation or as complex as self-transformation. Transformation is also used to describe a change in the nature of an entity. A transformation is a change in the relationship between variables. For example, if the independent variable increases, the dependent variable may increase, decrease or remain unchanged. Now, let’s discuss the big picture. Hopefully you understand how this matrix relates to changes in the controller and how those changes result in movement of the robot in the real world. A mapping function translates actions taken in the virtual world to movements of the controller. We arrived at the mapping function using linear approximation, which means that it is a good approximation only when the change in angle and distance between obiects is small. If you took a very large value of delta L, the prediction of this matrix would not be accurate. However, as long as delta L and delta theta are small, this approximation is accurate. If you took a very large value of delta L, the prediction of this matrix would not be accurate. However, as long as delta L and delta theta are small, this approximation is accurate. It provides an accurate approximation of a complicated problem. So, here is the explanation of the big picture. OK. So this thing with the robot arm-the controller and the real world – is an example of a transformation in mathematics. In transformation, we begin with some variables, L and theta, for example. This information then determines other variables, xy or whatever. This is called a transformation. A transformation is similar to a function, where x is dependent on L and theta, y is dependent on L and theta, etc. This is like a group of different functions, but they all work together to tell a coherent story about where the tip of the finger is. Those are some complicated topics. We’ll move on to linear approximation. It states that, if we consider only small changes in L and theta, then it is a lot simpler. In this case, the small changes in L and theta become small changes in x and y. This is a much simpler transformation. It’s called a matrix. Also known as a linear transformation. One reason that matrices are important is that any transformation can be modeled by a matrix, as long as we assume small variations in the variables. The concept of transformation is one that arises in many areas of science and engineering. A transformation is a change from one form, state, or place to another. There are many types of transformations, and they happen everywhere. It also occurs frequently in the field of machine learning. In the math department, we periodically send emails to people in different disciplines and ask them what is important for multi-variable calculus, or whatever class in their discipline that we should teach. I received an email from the professor who teaches the machine learning class, who said two things: gradients and understanding them. I plan over spring break to read some of the machine learning notes and try to understand why. I’ll tell you about it a little bit if we have time.
