# Determine whether T is a linear transformation. T:P2→P2 defined by T(a+bx+cx^{2})=(a+1)+(b+1)x+(c+1)x^{2}

Determine whether T is a linear transformation. $T:P2\to P2$ defined by
$T\left(a+bx+c{x}^{2}\right)=\left(a+1\right)+\left(b+1\right)x+\left(c+1\right){x}^{2}$
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Macsen Nixon

Here the mapping $T:P2-P2$ is defined by $\left(a+bx+c{x}^{2}\right)=\left(a+1\right)+\left(b+1\right)x+\left(c+1\right){x}^{2}$
Let V be a vector space and $T:V\to VT:V\to V$ is said to be a linear transformation if for $x,y\in V$ and $k1,k2\in K$ (the given field) we have $T\left(k1x+k2y\right)=k1T\left(x\right)+k2T\left(y\right).$
Here let $p\left(x\right)=1$ and $q\left(x\right)=x\in P2$. Then $T\left(1+2x\right)=1+1+\left(2+1\right)x=2+3x$ but $T\left(1\right)+2T\left(x\right)=1+1+2\left(1+1\right)x=2+4x.$
This shows that $T\left(1+2x\right)T\left(1+2x\right)$ is not equal to $T\left(1\right)+2T\left(x\right)$ and hence T is not a linear transformation. ​