How to find inverse mapping, if its invertible

Question

i know that to show invertibility, i have to prove that its either bijective or matrix is upper triangular. But i don't understand how to head start on this. Can anyone please help me with that?

Answer 1

to show invertibility, i have to prove that its either bijective or matrix is upper triangular.

Not all upper triangular matrices are invertible, so no. You could do something with matrices, and that would be a good second exercise, but you can really just directly see the answer to this.

Let’s get it of the way that, as sums of linear operators, these operators are linear.

The first one is obviously not invertible: $x^2$ is in the kernel. (That was the first thing I tried.)

Considering that the second one maps $a+bx +cx^2$ to $(a+b+c,b+2c,2c)$, it is clear that the kernel is zero and so the map is invertible.

The inverse is rather easy to see by inspection: solving for $a,b,c$ in $(X,Y,Z)=(a+b+c,b+2c,2c)$ yields $(X,Y,Z)\mapsto (X-Y+Z/2)+(Y-Z)x+(Z/2)x^2$ as the inverse image.