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T(a

_{0}+ a

_{1}t+a

_{2}t

^{2}) = 3a

_{0}+ (5a

_{0}- 2a

_{1})t + (4a

_{1}+ a

_{2})t

^{2}

is linear.Find the matrix representation of T relative to the basis B = {1,t,t

^{2}}

My book says to first compute the images of the basis vector. This is the point where I'm stuck at because I'm not sure how the books arrives at the images:

T(b1) = T(1) = 3+5t

T(b2) = T(t) = -2t+4t

^{2}

T(b3) = T(t

^{2}) = t

^{2}

Where are these results coming from?

I don't understand where 1 is supposed to go to solve for T(1). I guess its the notation that is throwing me off. Usually when solving for a transformation, it has something such as T(x) = x^2, and you solve the transformation by substituting the value of the input for x. But now my input is 1 for an entire expression (a

_{0}+ a

_{1}t+a

_{2}t

^{2})