 # Determine whether the set equipped with the given operations is a vector space. For those that Betsy Rhone 2022-01-04 Answered
Determine whether the set equipped with the given operations is a vector space.
For those that are not vector spaces identify the vector space axioms that fail.
The set of all pairs of real numbers of the form (x,0) with the standard operations on ${\mathbb{R}}^{2}$.
$\circ$ V is a vector space.
$\circ$ V is not a vector space, and Axiom 7,8, 9 fails to hold.
$\circ$ V is not a vector space, and Axioms 4 and 5 fail to hold.
$\circ$ V is not a vector space, and Axioms 2 and 3 fail to hold.
$\circ$ V is not a vector space, and Axiom 10 fails to hold.
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Asume $V=\left\{\left(x,0\right)\mid x\in R\right\}$
Let $V={R}^{2}$
Here we check subspace criteria for V
If it is satisfied then V is Vector space