matrix equations

Summary

Alternate representation of linear systems, as the product of a matrix and vector, and as a vector equation

• is the coefficient matrix
• is the variable vector
• is the constant vector

Row reduction of any matrix equation

• foundation for all reduction algorithms

row reducing gives as multiples of the columns of

Concept

Homogeneous linear systems are always consistent

visualise it as the origin/line formed by the intersection of planes(the linear equations) that passes through the origin

Combined augmented matrix

• linear systems on the same coefficient matrix

Application

Homongenous solution with infinitely many solutions

Relations between homogeneous and non-homogeneous systems

visualise a shifting of the plane to intersect the origin

Relationship between two non-homogeneous systems

On graphs, for modeling circuit/traffic flow problems

every connected graph reduces to a tree
cycles lead to dependent rows -> zero rows