Consider a non — homogeneous system of linear equation represent...
Consider a non-profit organization that focuses on providing education and resources to underprivileged children in developing countries. The organization's mission is to break the cycle of poverty by giving these children the opportunity to receive a quality education and gain the skills they need to succeed in life.
The organization partners with local schools and communities to identify children who are at risk of dropping out of school or who are unable to attend school due to financial constraints. They provide scholarships, school supplies, and access to educational resources such as books, computers, and internet facilities.
In addition to financial support, the organization also offers mentorship programs and extracurricular activities to enrich the children's educational experience. They organize workshops and seminars on topics like career guidance, personal development, and entrepreneurship to inspire and empower the children to dream big and achieve their goals.
The organization also recognizes the importance of involving parents and the community in the education process. They conduct workshops for parents to raise awareness about the value of education and provide guidance on how they can support their children's learning at home.
To measure the impact of their work, the organization regularly assesses the academic progress of the children they support and tracks their long-term outcomes. They also collect feedback from the children, parents, and teachers to continuously improve their programs and ensure they are effectively addressing the needs of the children.
To sustain their operations, the organization relies on donations from individuals, corporations, and grants from foundations. They also engage in fundraising activities such as charity runs, auctions, and crowdfunding campaigns.
Overall, this non-profit organization is dedicated to empowering underprivileged children through education, giving them the tools they need to break the cycle of poverty and build a brighter future for themselves and their communities.
Consider a non — homogeneous system of linear equation represent...
A
system of equations
is considered
overdetermined if there are more equations than unknowns.
Different cases of a overdetermined system , which are inconsistent and have no solution:
case 1: Consider the system of 3
equations
and 2 unknowns (
X and
Y), which is overdetermined because 3>2,
There is one solution for each pair of linear equations: for the first and second equations (0.2, −1.4), for the first and third (−2/3, 1/3), and for the second and third (1.5, 2.5). However, there is no solution that satisfies all three simultaneously.
case 2: A system of three linearly independent equations, three lines (two
parallel
), no solutions.
case 3: A system of three linearly independent equations, three lines (all parallel), no solution
In systems of linear equations, Li=ci for 1 ≤ i ≤ M, in variables X1, X2, ..., XN the equations are sometimes linearly dependent; in fact the number of linearly independent equations cannot exceed N+1. We have the following possible cases for an overdetermined system with N unknowns and M equations (M>N).
- M = N+1 and all M equations are
linearly independent
. This hull yields no solution. Example: x = 1, x = 2. - M > N but only K equations (K < M and K ≤ N+1) are linearly independent. There exist three possible sub-cases of this:
- K = N+1. This hull yields no solutions. Example: 2x = 2, x = 1, x = 2.
- K = N. This hull yields either a single solution or no solution, the latter occurring when the coefficient vector of one equation can be replicated by a weighted sum of the coefficient vectors of the other equations but that weighted sum applied to the constant terms of the other equations does not replicate the one equation's constant term. Example with one solution: 2x = 2, x = 1. Example with no solution: 2x + 2y = 2, x + y = 1, x + y = 3.
- K < N. This hull yields either infinitely many solutions or no solution, the latter occurring as in the previous sub-. Example with infinitely many solutions: 3x + 3y = 3, 2x + 2y = 2, x + y = 1. Example with no solution: 3x + 3y + 3z = 3, 2x + 2y + 2z = 2, x + y + z = 1, x + y + z = 4