Question: Let \(V\) be the vector space of polynomials with real coefficients and degree at most 2 . For each of the following linear transformations, determine whether they are one-to-one \(^{1}\) (so that \(\left.\operatorname{ker}(T)=\{0\}\right)\), onto (so that \(R(T)=V\) ), both, or neither.

1. \(T: \mathbb{R}^{3} \rightarrow V\) given by \(T(a, b, c)=a x^{2}+b x+c\)
2. \(T: \mathbb{R}^{3} \rightarrow V\) given by \(T(a, b, c)=a(x-1)+b(x+1)+c(x-1)(x+1)\)
3. \(T: \mathbb{R}^{2} \rightarrow V\) given by \(T(a, b)=a\left(x^{2}-2 x\right)+b(x-2)\)

Interpret each of your answers in terms of spanning sets and linear independence.

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