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If $$ A = \left[ \begin{matrix} 1 & -1 & 2 \\ -2 & 1 & -1 \\ 1 & 2 & 3 \\ \end{matrix} \right] $$ is the matrix representation of a linear transformation $$T :P_2(x) \to P_2(x)$$ with respect to the bases $\{1-x, x(1-x), x(1+x)\}$ and $\{1, 1+x, 1+ x^2\}$ then find T. What is the procedure to solve it? Also which book I can refer for more examples like this?

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    To begin with, do you know what it means for $A$ to be the matrix of $T$ with respect to a pair of bases? – 2017-01-13
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    @amd I know how to calculate matrix representation of a linear transformation if basis are given but not the other way around. Also, the book I am using (Schaum's Outline), there is not a single example of this type. Help! – 2017-01-13
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    The columns of the matrix of a transformation are the images of the basis vectors of the domain expressed relative to the basis of the codomain, i.e., it specifies a linear combination of the codomain basis vectors. So, the first column of $A$ tells us that $T[1-x]=1\cdot(1)-2\cdot(1+x)+1\cdot(1+x^2)=x^2-2x$, and so on. From that, you should be able to work out what $T$ does to the general polynomial $a+bx+cx^2$. Alternatively, you might convert $A$ to the standard basis and read the solution from that. – 2017-01-13

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Recall the the matrix of a transformation has as its columns the images of the domain basis vectors expressed relative to a basis of the codomain. Assuming that the first of the given bases is for the domain, this means that $$T[1-x] = 1\cdot(1)-2\cdot(1+x)+1\cdot(1+x^2) = x^2-2x.$$ The other two columns give you $T[x(1-x)]$ and $T[x(1+x)]$, respectively. Now, express the general second-degree polynomial $a+bx+cx^2$ as a linear combination of the domain basis polynomials, i.e., as $\alpha(1-x)+\beta x(1-x)+\gamma x(1+x)$ and use linearity of $T$: $$T[a+bx+cy]=\alpha T[1-x]+\beta T[x(1-x)]+\gamma T[x(1+x)].$$ Collect terms in the resulting polynomial, and you’re done.

Alternatively, apply a change of basis to $A$ to express it relative to the standard basis $(1,x,x^2)$ for both domain and codomain. On the codomain (output) side, we want to map from the given basis to the standard basis. This can be accomplished by multiplying by a matrix that has the given basis vectors as its columns. On the domain (input) side, we want to map from the standard basis to the given basis, which can be done by multiplying by the inverse of the matrix with the given basis vectors for its columns. The columns of the resulting matrix give you $T[1]$, $T[x]$ and $T[x^2]$, from which it is an easy matter to construct $T[a+bx+cx^2]$.