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Which of the following subsets of $\mathbb R^3$ are subspaces of $\mathbb R^3$? Just answer yes or no in each case.

(a) $\{(\alpha, \beta, \gamma) \mid \beta^2 = \alpha^2\}$;

(b) $\{(\alpha, \beta, \gamma) \mid \alpha^4 ≥ 0\}$;

(c) $\{(\alpha, \beta, \gamma) \mid α − 2β = γ\}$;

(d) $\{(\alpha, \beta, \gamma) \mid α + β = 3γ + 1\}$.

What I did : To verify a subset is a subspace, you need to show three things:

Show it is closed under addition. Show it is closed under scalar multiplication. Show that the vector 0 is in the subset.

a) Yes since if $α^2=β^2$ and $a^2=b^2$ then $(a+α)^2$ cannot be different from $(b+β)^2$

b) Yes since it is closed under scalar multiplication and addition

c) No, since it is not closed under scalar multiplication

d) No, since it is not closed under scalar multiplication

is that right?

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a) $(-1)^2 = 1^2$ and $1^2 = 1^2$, but $(-1 + 1)^2 \neq (1 + 1)^2$.

b) that is correct

c) correct

d) correct (and it doesn't contain 0)

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