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Problem 74: The set W of ordered triads $(a_1,a_2,0)$, where $a_1,a_2 \in F$ is a subspace of $V_3(F)$.

Problem 74: The set W of ordered triads $(a_1,a_2,0)$, where $a_1,a_2 \in F$ is a subspace of $V_3(F)$.

Solution:
$W = \{(a_1,a_2,0)|a_1,a_2 \in \F\}$.
Now let $u = (a_1,a_2,0)$ and $v = (b_1,b_2,0)$ for $a_1,a_2,b_1,b_2 \in \F$.Then
Closure under Addition:
$u+v = (a_1,a_2,0) + (b_1,b_2,0) = (a_1+b_1,a_2+b_2,0) = (x,y,0)$
if $x=a_1+b_1, y = a_2+b_2$
So, $u+v \in W$
Hence,W is closed under addition.
Closure under Scalar Multiplication:
Let $k \in \R$ and $u=(a_1,a_2,0) \in W$
Then, $ku = (ka_1,ka_2,0)$
which shows that $ku \in W$
Therefore W is a subspace of $V_3(F)$

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