Let 𝑊 be the set of vectors of the form 𝑎, 𝑎
2
, 𝑏 . Show
that 𝑊 is not a subspace of ℝ3
.

Question

Let 𝑊 be the set of vectors of the form 𝑎, 𝑎2, 𝑏 . Show&amp;nbsp;that 𝑊 is not a subspace of ℝ3.

nick1337 · Accepted Answer

To show that 𝑊 is not a subspace of ℝ3, we need to show that it violates at least one of the three properties that define a subspace.
Let's start by writing down the properties:
1. The zero vector,

0

, belongs to 𝑊.
2. 𝑊 is closed under vector addition. That is, if

𝐮

and

𝐯

are any vectors in 𝑊, then

𝐮 + 𝐯

is also in 𝑊.
3. 𝑊 is closed under scalar multiplication. That is, if

𝐮

is any vector in 𝑊 and

c

is any scalar, then

c 𝐮

is also in 𝑊.
Let's see if 𝑊 satisfies these properties.
1. The zero vector of 𝑊 is given by

(\begin{matrix} 0 \\ 0 \\ 0 \end{matrix})

. This vector is not of the form 𝑎, 𝑎

^{2}

, 𝑏, so it does not belong to 𝑊. Therefore, 𝑊 does not satisfy property 1 and cannot be a subspace of ℝ

^{3}

.
Since 𝑊 fails to satisfy one of the properties, it is not a subspace of ℝ

^{3}

.

Let 𝑊 be the set of vectors of

Answered question

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