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zohapmkoftid
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Homework Statement
If X_{0} and X_{1} are solutions to the homogeneous system of equation AX = 0, show that rX_{0} + sX_{1} is also a solution for any scalars r and s.
Thanks for help!
zohapmkoftid said:Homework Statement
If X_{0} and X_{1} are solutions to the homogeneous system of equation AX = 0, show that rX_{0} + sX_{1} is also a solution for any scalars r and s.
zohapmkoftid said:But how can we prove rX_{0} + sX_{1} = 0 from A(rX_{0} + sX_{1}) = 0
zohapmkoftid said:Homework Statement
If X_{0} and X_{1} are solutions to the homogeneous system of equation AX = 0, show that rX_{0} + sX_{1} is also a solution for any scalars r and s.
Thanks for help!
Homework Equations
The Attempt at a Solution
A homogeneous system is a set of linear equations in which all the terms have the same degree and are equal to zero. This means that all the unknown variables in the system have the same power.
To solve a homogeneous system, you can use the method of Gaussian elimination or matrix inversion. You can also use determinants and Cramer's rule to solve for the unknown variables.
Yes, a homogeneous system can have a unique solution if all the unknown variables are equal to zero. In this case, the system is called trivial.
If a homogeneous system has infinitely many solutions, it means that there are more unknown variables than equations, and the system is underdetermined. In this case, there is not enough information to determine a unique solution.
Homogeneous systems are commonly used in fields such as engineering, physics, and economics to model linear relationships between different variables. They can be used to solve optimization problems, analyze data, and make predictions.