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"let v = r 2 with the usual addition and scalar multiplication defined by k(u1, u2) = (ku1, 0). determine which of the five axioms of vector spaces involving scalar multiplication v satisfies and which fail. for the ones it satisfies, prove that it satisfies the axiom. for those that fail, show that it fails with a counterexample."

Respuesta :

LammettHash
Let [tex]\mathbf u\in\mathbb R^2[/tex], where

[tex]\mathbf u=(u_1,u_2)[/tex]

and let [tex]k\in\mathbb R[/tex] be any real constant.

Given this definition of scalar multiplication, we can see right away that there is no identity element [tex]e[/tex] such that

[tex]e\mathbf u=\mathbf u[/tex]

because

[tex]e\mathbf u=e(u_1,u_2)=(eu_1,0)\neq(u_1,u_2)=\mathbf u[/tex]

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