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Homogeneous systems

Predicting from rank counts which homogeneous systems admit non-trivial solutions, and building general solutions from basic solutions of the associated system.

Matrix Theory and Linear Algebra, Homogeneous systems

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2 exercises

Exercise 1.6.2

Which of the following homogeneous systems of linear equations have non-trivial solutions?

4 equations in 3 variables, rank 3.
3 equations in 4 variables, rank 3.
4 equations in 3 variables, rank 2.
3 equations in 4 variables, rank 2.

3.0

Exercise 1.6.3

My system of equations has a solution $(x,y,z)=(1,2,4)$ . The associated homogeneous system has basic solutions $(x,y,z)=(1,0,1)$ and $(x,y,z)=(0,1,-1)$ . What is the general solution of my system of equations?

3.0

Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.