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## Main Question or Discussion Point

**Problem**

Given a transformation T : P(t) -> (2t + 1)P(t) where P(t) ϵ P3

(a) Show that transformation is linear.

(b) Find the image of P(t) = 2 t^2 - 3 t^3

(c) Find the matrix of T relative to the standard basis ε = {1, t, t^2, t^3}

(d) Find the matrix of T relative to the basis β1 = {1, (1-t), (1-t)^2, (1-t)^3}

(e) Suppose [x]β1 = a column vector of 5 rows in which all entries = 2

find [x]ε

**Attempt**

For part a, should I show that the transformation of P(t) + transformation of Q(t) = transformation of P(t) + Q(t) ?

For part b, do I simply multiply P(t) by (2t+1) ?

I have no idea about parts c, d, e

Note: This is not a homework problem. In fact, I finished schooling years back. I need answers to these, with explanations please, to help somebody else. This thing was never in my syllabus. I had matrix but nothing related to basis.

So, could you explain to me or refer me to some web page from where I can learn this stuff? Thank you.