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Chapter 2: Q2.8-4Q (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.)

4.

Short Answer

Expert verified

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

01

State the condition for a subspace

A subspaceof \({\mathbb{R}^n}\) is any set Hin \({\mathbb{R}^n}\) that has three properties:

  1. The zero vector is in H.
  2. For each u and v in H, the sum \[{\mathop{\rm u}\nolimits} + v\] is in H.
  3. 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.

02

Draw the diagram with vectors in H

Draw the diagram with vectors in H. The scalar multiple of the vector is not in H, as shown below.

03

Explain that 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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