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Let V = R3. Show that V with the given operations for and is not a vector space. Clearly explain what goes wrong in terms of at least one of the axioms for vector spaces. C1 C2 T1+ 2 +5 21 L222 15 and T1 cc1 21 CZ1
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Answer #1

Given that V-(5-5-%)/4 e R Define for any α, β e Vwhere α-(xi, xa, x), β-(x,y2J3) the vector addition is defined by α田β-(xǐ両両, (M», у-(4+M +5両+½ +10丐切が the scalar multiplication by cOa=e(而ふろ)= (cy,cxa.c%)where c e R r r Considera o(α β)-a0( ,5-%),(yi,ya») ) anda@a@a@β-a@(4,x ,x3)r +a@(M.yaJ3f -(a, ax2 , a ) +(ay,ay,,ay,) Therefore a 0(α β) * a 0 α a Opror all a e R Hence is not a vector space with these operations.

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