“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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“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
“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
“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.
Carbon Neutral Economies
“How can an economy not contribute to climate change?”
Traditionally, economies have grown in tandem with carbon emissions. But with the advent of modern-day extreme climate change this is no longer feasible. To solve this, nations will have to adopt Carbon Neutral Economies in which all sectors have slashed their emissions to zero. The transition to a low carbon economy could yield enormous economic growth as well as inequality easing. Multiple governments around the world are committed to this action, with California’s 2045 and the EU’s 2050 initiatives as examples.
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“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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“What is an efficient method for pole placement in control systems?”
Pole Placement is an excellent way to stabilize a control system. But sometimes we may want an extra efficient way to do this, This is where we can apply Ackermann’s Formula, which automates it nicely for us.