Nibcode Solutions Blog

Vector spaces are one of the key subjects of linear algebra, and their theory has found application in mathematics, engineering, physics, chemistry, biology, the social sciences, and other areas. The theory, basically, consists in generalizing the familiar ideas of geometrical vectors of calculus to vectors of any size, but it provides an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. The beauty of vector spaces theory can be found in every problem, where many of them, are just the appropriate linear-algebraic notion of very well known problems like solving systems of linear equations.

The problem of this week is related to spanning sets and bases, two key concepts from vector spaces you should master. The solution, as usual in most of Linear Algebra problems, uses basic concepts from matrices and their operations.

One of the great potentialities of Linear Algebra Decoded is the possibility to generate problems where the coefficients are integers, and whose solution meets some given restrictions, so it allows students to generate problems covering different scenarios they can try, and in the same way it allows professors to generate problems that are convenient to use in exams. In this article we will explain how to use Linear Algebra Decoded to generate invertible matrices with integer coefficients where the inverse matrix has also integer coefficients.

Systems of linear equations lie at the heart of linear algebra, and they are used to solve practical problems in many fields of study. We can find examples in biology, economics and electronics, to cite only a few examples, where the solution to several problems is reduced to solve a system of linear equations.

So, this week, the problem will be related to this topic, where you should prove you understand the basic concepts when finding the solution of a system of linear equations.

This problem of the week is about the inverse of a square matrix, where you should need to know some basic properties and how to calculate it.

Inverse matrices are useful for a variety of fields, such as:

• Cryptography, to decrypt a message that was encrypted by using an invertible matrix.
• Image processing, where several transformations can be expressed as matrix multiplication, and to reverse those transformation, you just need multiply by the inverse.
• Solving consistent independent systems of linear equations.

When it comes to facing an exam, we all want to get good results, and although some people enjoy that moment, the reality is that for most it generates great tension.

From my personal experience, first as a student and then as a teacher, I would like to share some tips that will help you improve your results when you are faced with a demanding and laborious exam, such as those for Linear Algebra, although in general, these tips are valid for any exam.