Category: Mathematics

Climate Change Modeling

Climate Change Modeling

Climate Change Modeling

12/14/18

“How can we create models for the future climate?”

 

We seem to hear about the predictions of the future climate all the time in the media. Whether it be a warning if we go 2 degrees centigrade above pre-industrial levels or the ever-increasing rate of global warming. But how do we actually obtain this information? Well, people from a variety of different academic disciplines come together to make mathematical models that take in a variety of complex terrestrial processes and their interactions with each other and then layer it on to a physical model of the earth to produce a simulation of what will happen. This is known as Climate Change Modeling and is one of the most important tools that humans have to make policy goals.

 

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Column Space

Column Space

Column Space

12/11/18

“What is the space spanned by the linearly independent columns of a matrix?”

 

Matrices are likely to have at least a few linearly independent columns. The span of this space will result in a vector space known as a Column Space. The dimension of the column space is equivalent to the number of linearly independent vectors.

 

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Onto Functions

Onto Functions

Onto Functions

12/10/18

“What happens when all of the mappings in a linear transformation have at least one output?”

 

Sets of numbers can be changed around in linear transformations. And if every number maps onto at least one other number, then the function is Onto.

 

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Vector Spaces

Vector Spaces

Vector Spaces

12/09/18

“How can we represent a set of vectors?”

 

Vectors are one of the most important aspects of Linear Algebra. But in order to have a true set of vectors, they must be closed under vector addition and scalar multiplication. This is known as a Vector Space and is vital for a true understanding of Linear Algebra.

 

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Block Diagrams for Controls Systems

Block Diagrams for Controls Systems

Block Diagrams for Controls Systems

12/08/18

“How can we represent control process graphically?”

 

Control systems are a bedrock of modern day technology. However, sometimes it would be easier to see their processes if visualized. This can be done by using Block Diagrams, which shows each component of the control system in its natural order.

The LQR Function

The LQR Function

The LQR Function

12/07/18

“How can we create a control function with easy inputs?”

 

Modern control systems rely on matrix operations. There are multiple ways to create optimal varieties of this. One such way is to create a cost function which assigns different penalties to each input. This is known as the LQR Function and is implemented in a multitude of modern control systems around the world.

 

Matrix Transposition

Matrix Transposition

Matrix Transposition

12/05/18

“How can we flip rows and columns of a matrix?”

 

When doing Linear Algebra, there are times where we may want to flip the rows and columns. This can be accomplished through Matrix Transposition. A matrix that has undergone transposition is known as a transposed matrix. If we multiply the transposed matrix by the original matrix then we arrive at the identity matrix.

 

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The Span of a Matrix

The Span of a Matrix

The Span of a Matrix

12/03/18

“How can we represent all possible combinations of a matrix?”

 

Matrices are great for representing groups of numbers. But sometimes we would like to see all possible results. We can do this using the Span of a Matrix, using weighted values for each part.