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Chapter 3: Q1E (page 131)

Which of the sets W in Exercises 1 through 3 are subspaces of鈩�3?

1.W={xyz:x+y+z=1}

Short Answer

Expert verified

W is not a subspace of 鈩�3.

Step by step solution

01

Step 1: Conditions

All three conditions must hold in order for W to be a subspace in鈩�3these are:

1) W contains the zero vector in鈩�3.

2)Ifw鈫�1andw鈫�2arecontainedinW,thensoisw鈫�1+w鈫�2

3) Ifw1鈫� is contained in W, then so is kw1鈫� where k is a scalar.

02

Checking condition 1

Now consider,w=xyz:x+y+Z=1.We will check condition 1.

000:0+0+0鈮�1

Thus, W fails condition 1 and is not a subspace of 鈩�3.

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