Linear Algebra is one of the most important branches of mathematics, as it has applications in a wide range of fields including engineering, computer science, and physics.
Purpose of Linear Algebra
The main purpose of linear algebra is to understand and describe relationships between vectors (i.e., objects with magnitude and direction) and linear transformations.
Linear equations are equations involving variables that can be expressed in the form ax+by=c where a, b and c are known constants.
Linear algebra helps us solve such problems without needing to find an exact solution for all values of x or y individually; instead we solve for all possible solutions using matrix methods which help immensely when dealing with large number of equations or unknowns.
Graphing Data
Another major use case for linear algebra is graphing data – plotting points on a coordinate plane to identify trends or correlations between two or more datasets by analyzing their relation through vectors within space (or higher dimensions).
For example, if we want to predict stock prices over time then by studying the results from different datasets through matrices we can obtain useful information about how price changes interact with other factors like economic growth rate, inflation, etc., which can then be used to predict future stock movements.
Where Linear Algebra Has Been Applied
Beyond that, linear algebra has been applied extensively in areas like:
- control theory for optimization leading researchers closer towards finding optimal solutions quickly
- game theory where matrices are used extensively in predicting win/loss scenarios
- quantum mechanics which relies heavily on matrix operations to study quantum phenomena at its core level
- image processing – very useful in medical sciences especially MRI images etc.
- network analysis – efficient way of finding shortest paths from start node (vertex) to end node while circumventing obstacles etc.
- pattern recognition – classification based on certain criteria as determined by patterns stored inside matrices/vectors upon inputs e.g., facial recognition systems used widely today employ this technique substantially too
