Please explain your answer, thanks Decide whether each map is an isomorphism (if

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Please explain your answer, thanks

Decide whether each map is an isomorphism (if it is an isomorphism, then prove it and if it isn't then state a condition that it fails to satisfy). f: M2 times 2 rightarrow p3 given by (a b c d) c + (d + c)x + (b + a)x2 + ax3

Explanation / Answer

it will be an isomorphism if it is one-one . so, we only have to show that this is one-one we have for two x1 and x2 c+ (d+c)x1 + (b+a)x1^2 + a x1^3 = c+ (d+c)x2 + (b+a)x2^2 + a x2^3 => (d+c)(x1-x2) + (b+a)(x1^2 - X2) + a (x1^3 - x2^3) =0 => (d+c) (x1 - x2) + (b+a)(x1-x2)(x1 + x2) + a(x1-x2) ( x1^2 + x2^2 +x1x2) =0 => (x1 - x2)[ (d+c) + (b+a)(x1 + x2) + a( x1^2 + x2^2 +x1x2) ] =0 => x1 - x2 =0 => x1 = x2 hence this is one-one and hence an isomorphism

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