If the linear system a1 x+b1 y+c1 z=0 a2 x-b2 y+c2 z=0 a3 x+b3 y-c3 z=0 has on

step 1 Hello there. So for this occasion we have this system of linear equations and we know that this

If the linear system a1 x+b1 y+c1 z=0 a2 x-b2...

If the linear system $$ \begin{array}{l} a_{1} x+b_{1} y+c_{1} z=0 \\ a_{2} x-b_{2} y+c_{2} z=0 \\ a_{3} x+b_{3} y-c_{3} z=0 \end{array} $$ has only the trivial solution, what can be said about the solutions of the following system? $$ \begin{array}{l} a_{1} x+b_{1} y+c_{1} z=3 \\ a_{2} x-b_{2} y+c_{2} z=7 \\ a_{3} x+b_{3} y-c_{3} z=11 \end{array} $$

If the linear system $$ \begin{array}{l} a_{1} x+b_{1} y+c_{1} z=0 \\ a_{2} x-b_{2} y+c_{2} z=0 \\
a_{3} x+b_{3} y-c_{3} z=0 \end{array} $$ has only the trivial solution, what can be said about the solutions of the following system? $$ \begin{array}{l} a_{1} x+b_{1} y+c_{1} z=3 \\
a_{2} x-b_{2} y+c_{2} z=7 \\ a_{3} x+b_{3} y-c_{3} z=11 \end{array} $$

Key Concepts

Homogeneous Linear Systems

A homogeneous linear system is one in which all of the constant terms are zero. Such systems always have at least the trivial solution (all variables equal to zero), and if this is the only solution, it indicates that the system's equations are linearly independent.

Trivial Solution and Linear Independence

The fact that the homogeneous system has only the trivial solution means that the set of equations has no nontrivial linear combinations producing the zero vector, which is a direct indicator of the linear independence of the equation coefficients. This, in turn, shows that the associated coefficient matrix is invertible.

Invertible Matrices and Unique Solutions

An invertible matrix, characterized by a non-zero determinant, guarantees that a linear system has a unique solution. When the coefficient matrix is invertible, any non-homogeneous system with that matrix will yield exactly one solution, obtained by applying the inverse matrix to the constants vector.

Non-Homogeneous Linear Systems

A non-homogeneous linear system is one that includes nonzero constants on the right-hand side. If the coefficient matrix of such a system is invertible, the system will have exactly one unique solution. This concept is crucial for understanding how the properties of the homogeneous system extend to the non-homogeneous case.

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