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}

emancipezN 2020-12-28 Answered
Determine whether T is a linear transformation. T:P2P2 defined by
T(a+bx+cx2)=(a+1)+(b+1)x+(c+1)x2
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Expert Answer

Macsen Nixon
Answered 2020-12-29 Author has 117 answers

Here the mapping T:P2P2 is defined by (a+bx+cx2)=(a+1)+(b+1)x+(c+1)x2
Let V be a vector space and T:VVT:VV is said to be a linear transformation if for x,yV and k1,k2K (the given field) we have T(k1x+k2y)=k1T(x)+k2T(y).
Here let p(x)=1 and q(x)=xP2. Then T(1+2x)=1+1+(2+1)x=2+3x but T(1)+2T(x)=1+1+2(1+1)x=2+4x.
This shows that T(1+2x)T(1+2x) is not equal to T(1)+2T(x) and hence T is not a linear transformation. ​

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