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Decide, if linear mapping $f: M_{22}(R)\to P_3(x,R) $ defined as

$ f $$\left(\begin{pmatrix}a&b\\c&d\\\end{pmatrix}\right)$$ = a + d + (c + 2b)x^2 - x^3$

is linear.

Well, I do not know what to do if we have mixed matrices and polynoms. I tried to put that matrix as x in that mapping ... but it's nonsense.

Mapping is linear if: $ f(u+v) = f(u) + f(v) $ and $ $ $f(ru) = rf(u) $ .

Thanks for helping.

1 Answers 1

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Notice that the zero element in $M_{22}( R )$ is the zero matrix, i.e $\mathbf 0.$ Notice that: $$f(\mathbf 0) = -x^3 \neq 0.$$ If we have a linear map $f: V\to W,$ then it has to hold $f(\mathbf 0_V) = \mathbf 0_W.$

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    I see, but I don't even understand that transformation. How could I transform any matrix into polynom? Basic question but unfortunately I have no intuition for linear algebra. But thank you. – 2017-01-28
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    Why is it that strange? You can consider the mapping: $(a,b,c,d) \mapsto (a+d) + (c+2b)x^2 - x^3,$ i.e. every ordered 4-tuple maps to a 3rd degree polynomial. – 2017-01-28