Calculate the core of a matrix

What you need:

• Basics in matrix calculations

Matrix and linear mapping - the connection

• A matrix is ​​initially nothing more than an ordered collection of (mostly) Counting. The arrangement takes place in rows and columns, so you speak of an m x n matrix with m rows and n columns.
• Matrices have a variety of uses. For example, they can represent systems of linear equations. But matrices also play a role in the area of ​​mathematical mapping (rotations, shifts, reflections).
• With a matrix you can represent a linear mapping between two vector spaces, i.e. between sets that contain vectors. In the simplest case, a matrix maps vectors of three-dimensional space onto other vectors there, for example as a reflection on a plane.
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• You compute the image of any vector by dividing the matrix with this multiply.

Image, core and set of fixed points - simply explained

• Mathematicians are familiar with three important, fundamental terms for linear mappings, which are represented as a matrix, namely image, core and set of fixed points in the map or the matrix.


Matrix problems - this is how you multiply two matrices

Multiplying two matrices is - if you follow the rules for it - actually ...

• The image of a matrix consists of the vectors that you generate when you apply the matrix to all possible vectors in your original vector space. In a way, this picture is similar to the set of values ​​of a function.
• The core of a matrix is ​​the set of all vectors (or points) that are mapped from this matrix to the zero vector. If A is the matrix, then calculate the vector x you are looking for using the equation A * x = 0. Here, 0 symbolizes the zero vector, which cannot be represented here with an arrow. The kernel of a matrix is ​​therefore generally a subset of the original vector space.
• The set of fixed points of a matrix is ​​the set of vectors that are mapped onto itself by matrix A. Put simply, you can apply the mapping to this set of vectors and everything stays the same.

Illuminate the theory - calculate examples

Such parts of the theory are gray and often opaque. For this reason, some basic examples are intended to illuminate the terms in this section:

• The simplest figure is the so-called. Zero mapping in which all points or Vectors of the R³ can be mapped onto the zero vector. Associated with this figure is a 3 x 3 matrix that contains only zeros. The image set consists of a single element, namely the zero vector. The core of the matrix is ​​the complete R³, because all vectors are mapped to zero. The set of fixed points is also clear, it only consists of the zero vector.
• The so-called identical mapping (also called identity) has the identity matrix as matrix, for example E₃ in three-dimensional space. The image set is the complete R³, The core is only the zero vector and the set of fixed points is also the complete R³.
• If you want to calculate the kernel for an arbitrary matrix A, your work boils down to solving a linear system of equations. Because as a condition you have A * x = 0. If one calculates the left side, then three result for the three-dimensional case, for example Equations with the three coordinates of the vector x as unknowns.