Final answer:
An example system of five equations with five unknowns with a rank of 2 is one where only two of the equations are linearly independent; the other equations are multiples or combinations of these two independent equations. Equation 1 (x + 2y = 4) and Equation 4 (x - 2y + 3z = 1) can serve as the basis for the linearly independent set. The system is underdetermined, with an infinite number of solutions defined by parameters.
Step-by-step explanation:
To create a system of five equations and five unknowns that has a rank of 2, we can construct a system where only two of the equations are linearly independent, and the remaining equations are multiples or combinations of those two. An example of such a system is:
x + 2y = 4 (Equation 1)
2x + 4y = 8 (Equation 2)
3x + 6y = 12 (Equation 3)
x - 2y + 3z = 1 (Equation 4)
2x - 4y + 6z = 2 (Equation 5)
Here, Equations 2 and 3 are simply multiples of Equation 1, and thus they are not considered to add any new information or increase the rank. Likewise, Equation 5 is a multiple of Equation 4. Therefore, the rank of this system is 2 because there are only two linearly independent equations, which are Equations 1 and 4.
Equations 1 and 4 are the basis for the linearly independent set, while the other equations do not contribute to increasing the rank of the system. Solving this system involves using algebraic methods to find solutions for the unknowns, but since the system is underdetermined (more unknowns than independent equations), there will be an infinite number of solutions, defined by a set of parameters.