Holly Guerrero

2022-01-04

Problem 2:
If $V={R}^{3}$ is a vector space and let H be a subset of V and is defined as $H=\left\{\left(a,b,c\right):{c}^{2}+{b}^{2}=0,a\ge 0\right\}$. Show that H is not subspace of vector space
Problem 3
Let $V={R}^{3}$ be a vector space and let W be a subset of V, where $W=\left\{\left(a,b,c\right):{a}^{2}={b}^{2}\right\}$. Determine whether W is a subspace of vector space or not.

braodagxj

Expert

Problem 2:
$V={R}^{3}$
$H=\left\{\left(a,b,c\right):{c}^{2}+{b}^{2}=0,a\ge 0\right\}$
At $x=\left(1,0,0\right)\in H$
But $\alpha =-2$ is scalar in R
$\alpha x=\left(-2\right)\left(1,0,0\right)$
$=\left(-2,0,0\right)\notin H\left(-2<0\right)$
Not subspace of vector space V

Piosellisf

Expert

Problem 3:
$V={R}^{3}$
$W=\left\{\left(a,b,c\right):{a}^{2}={b}^{2}\right\}$
At $x\left(1,-1,0\right)\in H$
At $y\left(1,1,0\right)\in H$
But $x+y=\left(2,0,0\right)\notin H$
$\therefore {2}^{2}\ne {0}^{2}$
Not subspace of vector space V

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