Let v be the set of vectors in r2 with the following definition of addition and scalar multiplication: addition [x1 x2] [y1 y2] = [0 x2 + y2] scalar multiplication: alpha [x1 x2] = [ax1 ax2] determine which of the vector space axioms are satisfied.

Given Set is not a vector space as Axiom 4 and 8 are not satisfied.

Step-by-step explanation:

Given:

V is Vector  in R².

Definition of addition :

Definition of multiplication :

We need to check which axioms of the vector space satisfied by V.

Let,

and α , β are scalars.

Axiom 1).

Axiom 2).

⇒ u + v = v + u

Axiom 3).

⇒ ( u + v ) + w = u + ( v + w )

Axiom 4).

⇒ u + 0 ≠ u

Axiom 5).

⇒ u + (-u) = 0

Axiom 6).

Axiom 7).

⇒ α ( u + v ) = αu + αv

Axiom 8).

⇒ ( α + β )u ≠ αu + βu

Axiom 9).

⇒ ( αβ )u = α( βu )

Axiom 10).

⇒ 1 . u = u

Therefore, Given Set is not a vector space as Axiom 4 and 8 are not satisfied.

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