The set of all first degree polynomial functions ax+b, a not equal to zero, whose graphs pass through the origin with the standard operations is not a vector space. State all of the axioms that fail and give a counterexample for each.
The Correct Answer and Explanation is :
The set of all first-degree polynomial functions of the form ( ax + b ), where ( a \neq 0 ) and the graph passes through the origin, does not form a vector space. Let’s go through the axioms of a vector space and identify which ones fail.
1. Axiom 1: Closure under addition
A vector space must be closed under addition. This means if ( f(x) = ax + b ) and ( g(x) = cx + d ) are in the set, their sum ( f(x) + g(x) = (a + c)x + (b + d) ) must also be in the set.
Failure:
The sum of two functions that pass through the origin may not pass through the origin. For example, ( f(x) = 2x ) and ( g(x) = 3x ) both pass through the origin, but their sum ( (2x + 3x) = 5x ) does not pass through the origin (as ( b \neq 0 )).
Counterexample:
Let ( f(x) = 2x ) and ( g(x) = 3x ). Their sum is ( f(x) + g(x) = 5x ), which does not pass through the origin.
2. Axiom 2: Closure under scalar multiplication
A vector space must be closed under scalar multiplication. If ( f(x) = ax + b ) is in the set, and ( k ) is a scalar, then ( k \cdot f(x) = k \cdot (ax + b) = k \cdot ax + k \cdot b ) must be in the set.
Failure:
The scalar multiple of a function that passes through the origin may not pass through the origin. For example, ( f(x) = 2x ), and multiplying by a scalar ( k = 3 ) gives ( 3f(x) = 6x ), which still passes through the origin, but multiplying by a negative scalar (say, ( k = -1 )) could result in a function that doesn’t pass through the origin, depending on the value of ( b ).
Counterexample:
Let ( f(x) = 2x ) and ( k = -1 ). Then ( k \cdot f(x) = -2x ), which still passes through the origin, but if we add a constant term like ( b ), the result may not.
Conclusion:
The primary failure is closure under addition, as the sum of two polynomials that pass through the origin may result in a polynomial that does not pass through the origin. Therefore, the set does not form a vector space.
Now I will generate an image illustrating these examples and axioms.
Here is a graph illustrating the examples for the vector space axioms that fail:
- The first line passes through the origin.
- The second line represents the sum of two lines that both pass through the origin, but the sum does not pass through the origin (failure of closure under addition).
- The third line shows a scalar multiple of a line that passes through the origin, but may not always pass through the origin with other operations (failure of closure under scalar multiplication).
This image should help visualize the failures of the axioms. Let me know if you need further explanation!
