Please answer the questions: Let V = R2 but with addition and multiplication def

Question

Please answer the questions:

Let V = R2 but with addition and multiplication defined by (x1,x2) (y1,y2) = (x1 + y1, 0) C (x1,x2) = (cx1,cx2) What is the first vector space axiom that fails? . Let V=R2 but with addition and multiplication defined by (x1,x2) (y1,y2) = (x1 + y1, x2 + y2) C (x1,x2) = (0,0) What is the first vector space axiom that fails? . Let V = M2x2 but with addition and multiplication defined by A B = AB C A = cA What is the first vector space axiom that fails? . Let V = P1 but with addition defined by (a0 + a1x) (b0 + b1x) = a1 + b1 + (ao + bo)x. What is the first vector space axiom that fails?

Explanation / Answer

V1. The failed axiom is distributive axiom i.e.

For every (x1,x2),(y1,y2) in V and real numbers c,d, we have: c((x1,x2)+(y1,y2)) = c(x1,x2) + c(y1,y2)

V2.The failed axiom is one multiplicative axiom i.e

For every (x1,x2) in V, we have 1*(x1,x2) = (x1,x2)

V3.The failed axiom is one additive axiom i.e

For every A,B in V, we have A+B = B+A

V4. The failed axiom is one additive axiom i.e

For every (a0+a1x), (b0+b1x), (c0+c1x) in V, we have ((a0+a1x)+(b0+b1x))+(c0+c1x) = (a0+a1x)+((b0+b1x)+(c0+c1x)).

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