Question

Why does a homogenous linear system in n unknowns whose corresponding augmented matrix with rref containg n leading 1s, then the linear system onlyhas the trivial solution?

Answer 1

A homogeneous system always has a nontrivial solution if the number of equations is less than the number of unknowns.

What is homogenous linear system ?

A homogeneous system of linear equations where each equation's constant term is equal to 0. Therefore, there are no constant terms in any equation for such systems. There might be one solution or infinitely many for a homogeneous linear system. But there is always at least one answer.

There are no constant terms in a homogeneous system of linear equations, which is a linear equation system.

There are two different kinds of solutions for a homogeneous system: trivial solutions and nontrivial solutions. Since there is no constant term in homogeneous systems, the simple solution is (x₁, x₂, ..., xₙ) = (0, 0,..., 0), which is an evident solution to the problem (the most obvious solution). For instance, the solution (x, y, z) = is simple for the system created by the three equations x + y + z = 0, y - z = 0, and x + 2y = 0. (0, 0, 0).

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