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Magazine Chapter 2: Q2.8-1Q (page 93)
Exercises 1-4 display sets in \({\mathbb{R}^2}\). Assume the sets include the bounding lines. In each case, give a specific reason why the set H is not a subspace of \({\mathbb{R}^2}\). (For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.)
1.
Short Answer
Set H is closed under addition but not under multiplication by a negative scalar. Thus,itis not a subspace of \({\mathbb{R}^2}\).
Step by step solution
State the condition for a subspace
A subspaceof \({\mathbb{R}^n}\) is any set H in \({\mathbb{R}^n}\) that has three properties:
- The zero vector is in H.
- For each u and v in H, the sum \[{\mathop{\rm u}\nolimits} + v\] is in H.
- For each u in H and each scalar c, the vector \[c{\mathop{\rm u}\nolimits} \] is in H.
Asubspace is closed under addition and scalar multiplication.
Draw the diagram with vectors in H
Draw the diagram with vector in H. The scalar multiple of the vector is not in H, as shown below.
Explain why set H is not a subspace of \({\mathbb{R}^2}\)
Set H is closed under addition but not under multiplication by a negative scalar.
Thus, set His not a subspace of \({\mathbb{R}^2}\).
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