Basic Linear Algebra — System of Equations
The system of equations is one basic concept for understanding or solving some real world problems. Let us see how we can do that
The linear equations are divided into Complete, Redundant and Contradictory equations.
Complete Linear Equation — A complete system of linear equations has exactly as many independent equations as there are variables, providing a unique solution.
Real world example —
You are planning a shopping trip and want to budget for two items: shirts and pants. You have a total budget and want to determine how many of each item you can buy.
- Cost of each shirt: $20
- Cost of each pair of pants: $20
- Total budget: $120
You also want 2 pants more than the shirts. Here, you have to find how many pants and shirts you can buy within your budget.
Imagine you want to buy x number of pants and y number of shirts.
20 x + 20 y = 120
x = y + 2We can get these 2 equations with the constraints. The first equation says the budge t should be within 120 and second condition says number of shirts should be 2 less than pants.
Here there are 2 equations and these 2 has useful information and it is not repeating the same information. With the 2 equations we can solve by replacing x with y+2 in the first equation, we get number of shirts as 2 and number of pants as 4
Redundant Linear Equation — A redundant linear equation in a system is one that does not provide any new information because it is a linear combination of other equations in the system. This means that the redundant equation does not change the solution set of the system
Real world example —
Suppose you bought an apple and a banana and they cost $10. You bought 2 apples and 2 bananas cost $20. Here we can say apple as $8 and $2 as banana or $5 as apple and $5 as banana or any other value also satisfies. The both sentences gives the same information where the cost of banana and cost of apple cannot be inferred.
Contradictory Linear Equation — A contradictory system of linear equations is one where no set of values satisfies all the equations simultaneously. This occurs when the equations represent constraints or conditions that are logically impossible to meet at the same time
Real World Example —
You have to book a meeting room for 2 meetings and it should accommodate 2 timings. Meeting A is from 2PM — 4PM and Meeting B is from 3PM — 5 PM. Here the timings overlap and you cannot book the meeting room at the same time for both the meetings. It does not satisfy the conditions.
From the above examples, we say the complete equation which is not redundant/contradictory is a non-singular equation and the other 2 are singular equations.
Let us take it to next level. Find the below diagram
The first set of equations has infinitely many solutions and we cannot get any output. This is redundant as all the 3 equations gives the same output which cannot be concluded. As this is redundant, it is singular equations. In second case, we get c=5 from 1st and 2nd but c=3 from 2nd and 3rd so this is contradictory. This also leads to no solution and is singular. The third one has the all the equations which gives the same thing which is a+b+c=10. This is redundant and singular.
When we visualize these equations, we find something interesting. The equations which gives exactly one solution when visualized meets at one point and is non-singular. If the equations has infinite solutions then they are the same line and redundant. So it has infinitely many solutions. If 2 solutions are contradictory they are parallel and do not meet at all and they have no solutions. In both cases of redundant and contradictory, they are singular.
Hope this helps in recognizing the equations are singular / non-singular
While we say the equations are singular and non-singular, we can also make them
