From Variables to Matrices in Mathematics and Machine Learning

Big picture Transformation is a fundamental change in the nature of something. This may be assimple as an outward change, such as a physical transformation or as complex asself-transformation. Transformation is also used to describe a change in the nature of anentity.A transformation is a change in the relationship between variables. For example, if theindependent variable increases, the dependent variable may increase, decrease or remainunchanged. Now, let's discuss the big picture. Hopefully you understand how this matrix relates tochanges in the controller and how those changes result in movement of the robot in thereal world. A mapping function translates actions taken in the virtual world to movements of thecontroller. We arrived at the mapping function using linear approximation, which meansthat it is a good approximation only when the change in angle and distance betweenobjects 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. It provides an accurate approximation of a complicated problem. So, here is theexplanation of the big picture. OK. So this thing with the robot arm—the controller and thereal world – is an example of a transformation in mathematics. In transformation, we beginwith some variables, L and theta, for example. This information then determines othervariables, xy or whatever. This is called a transformation. A transformation is similar to a function, where x isdependent on L and theta, y is dependent on L and theta, etc. This is like a group ofdifferent functions, but they all work together to tell a coherent story about where the tip ofthe finger is. Those are some complicated topics. We'll move on to linear approximation. It states that, ifwe consider only small changes in L and theta, then it is a lot simpler. In this case, thesmall 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 lineartransformation. One reason that matrices are important is that any transformation can be modeled by amatrix, as long as we assume small variations in the variables. The concept of transformation is one that arises in many areas of science andengineering. A transformation is a change from one form, state, or place to another. Thereare many types of transformations, and they happen everywhere. It also occurs frequentlyin the field of machine learning. In the math department, we periodically send emails to people in different disciplines andask them what is important for multi-variable calculus, or whatever class in their disciplinethat we should teach.

I received an email from the professor who teaches the machine learning class, who saidtwo things: gradients and understanding them. I plan over spring break to read some of the machine learning notes and try to understandwhy. I'll tell you about it a little bit if we have time.