Math Problem Statement
Solution
Let's go through each part of the problem carefully.
Problem 7:
Let be the first quadrant in the -plane, i.e., This defines the set of all vectors in the first quadrant of the plane.
Part (a):
If and are in , is in ? Why?
Solution:
- Let and be two vectors in .
- This means that , and , .
- The sum of the vectors .
Since both and , we know . Similarly, . Therefore, is also in the first quadrant, which means it is in .
Conclusion: Yes, if and are in , then is also in .
Part (b):
Show that is not a vector space.
Solution:
A set of vectors forms a vector space if it satisfies the following conditions, among others:
- Closure under addition (which we verified in part (a)).
- Closure under scalar multiplication, meaning if and , then .
Let's test closure under scalar multiplication:
- Let (this vector is in the first quadrant since both components are positive).
- Now, take a scalar . Then .
This vector is not in because both components are negative (it lies in the third quadrant). Hence, is not closed under scalar multiplication.
Conclusion: is not a vector space because it is not closed under scalar multiplication. Specifically, multiplying a vector in by a negative scalar results in a vector that is not in .
Let me know if you'd like further details on any part of the solution. Here are some related questions to consider:
- Can you define other conditions necessary for a set to be a vector space?
- How would the definition of change if it included vectors from more quadrants of the plane?
- What role does scalar multiplication play in defining vector spaces?
- Is the set of all vectors in the first quadrant a subspace of ?
- Could the set form a vector space if restricted to specific operations or scalars?
Tip: Always check both closure under addition and scalar multiplication when verifying if a set forms a vector space.
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Math Problem Analysis
Mathematical Concepts
Vector Addition
Vector Spaces
Closure Property
Scalar Multiplication
Formulas
u + v = (u1 + v1, u2 + v2)
c * u = (c * u1, c * u2)
Theorems
Vector space axioms
Closure under addition and scalar multiplication
Suitable Grade Level
Undergraduate Mathematics
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