Question

Applications of Rank Theorem to Linear Equations Prompt Discussion In this discussion, a real-life problem using linear equations might be stated, without explicit mention of linear algebra terms such as matrix, subspace, and dimension. We can reformulate the problem in linear algebraic terms and then give a solution. The Rank Theorem is a powerful tool for processing information about systems of linear equations.

1. A scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will have a solution? Discuss.
2. Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution? Discuss.

Answer 1

A scientist assured of solutions for varying nonhomogeneous equations due to the existence of free variables. Not all solutions to a homogeneous system can be multiples of a single nonzero solution given the excess of variables over equations. Kirchhoff's Rules are a practical example of using linear equations for problem-solving.

Step-by-step explanation:

Applications of the Rank Theorem

Concerning the first inquiry, if a scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and discovers three free variables, they can be certain that a solution exists for any right side of the equations.

This is because the Rank Theorem indicates that the rank of the matrix plus the number of free variables equals the number of unknowns. Since there already are free variables, altering the right sides of the equations won't affect the existence of a solution; however, it will affect the solution set.

In response to the second inquiry, it is not possible for all solutions of a homogeneous system of ten linear equations in twelve variables to only be multiples of one fixed nonzero solution. The system has more variables than equations, which implies there must be at least two free variables leading to an infinite number of solutions that are not simply multiples of one another but span a multidimensional space.

Using Kirchhoff's Rules in circuit analysis involves generating a set of linear equations to solve for unknown currents, voltages, or resistances. If there are as many independent equations as unknowns, the problem can be solved. This practical application demonstrates the significance of the Rank Theorem in real-life scenarios.