A linear system whose equations are all homogeneous must be consistent.
1
This statement is true.
A linear system of equations is considered homogeneous if all the constants on the right side of the equations are zero. This means that the system can be written in the following form:
a1x1 + a2x2 + … + anxn = 0
b1x1 + b2x2 + … + bnxn = 0
…
z1x1 + z2x2 + … + znxn = 0
If we look at this system, we can see that the trivial solution (x1 = x2 = … = xn = 0) will always satisfy all the equations. This is because if we substitute all the x values by zero, the left side of each equation will become zero as well, which is equal to the right side of all the equations.
Therefore, since there is always a solution that satisfies all the equations, a homogeneous linear system must be consistent.
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