Create a bijective transformation that maps P2 to R^3? If so, create the map, or explain why this is not possible.

I'm looking at page 292 of the Linear Algebra and Its Applications 5th Edition Textbook by David C. Lay. Could someone break down Example 2 some more for me? I don't understand how T(1) = 0, T(t) = 1, and T(t^2) = 2t or which function we should look t for the mapping. The entire part a section of that Example 2 problem I need help with.

why no matrix exists for one-to-one linear transformation from R^4 to R^3

Define identity transformation.

Determine whether the function is a linear transformation. T: P₂ → P₂, Tao + a₁x + a₂x²) linear transformation O not a linear transformation = (ao + a₁ + a₂) + (a₁ + a₂)x + a2x²

Please help

Name the type of transformation

What is a transformation in which a figure is turned about a fixed point?

Define the function Q : M2x2 M2x2 by Q(A) = A – AT. Show that Q is a linear transformation.

Please describe using text keyboard (NOT HANDWRITING), what is meant by “The Kernel,” and “The Range” of a linear transformation. Put simply, what are they?