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## Homework Statement

Determine whether ##W = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : a_1 = 3a_3,~ a_3 = -a_2 \}## is a subspace of ##\mathbb{R}^3##.

## Homework Equations

## The Attempt at a Solution

To show that a subset of vector space is a subspace we need to show three things: 1) That the zero vector of R^3 is in W. 2) That W is closed under vector addition. 3) That W is closed under scalar multiplication.

1) if ##a_2 = 0## then ##a_1 = 0,~a_3 = 0##, so the zero vector is in W.

2) I'm not exactly sure how to clearly show this one. Here is my attempt: ##(a_1, a_2, a_3)+ (b_1, b_2, b_3) = (a_1 + b_1, a_2 + b_2, a_3 + b_3) = (3(a_2 + b_2), a_2 + b_2, -(a_2 + b_2))##, which is of the form of the vector defined in W.

3) Not exactly sure how to show this one either, but here is my attempt: ##c(a_1, a_2, a_3) = (ca_1, ca_2, ca_3) = (3(ca_2), ca_2, -(ca_2))##, which is of the form of the vectors in W.

Thus, W is a subspace of R^3

Is this a correct proof? Am I doing 2) and 3) right or is there a better way?