Determine whether the linear transformation T is one-to-one

[hannahlu92]

Determine whether the linear transformation T is one-to-one

a) T:P2 --> P3, where T(a+a1x+a2x^2)=x(a+a1x+a2x^2)

b) T:P2 --> P2, where T(p(x))=p(x+1)

I'm having difficulty because my teacher never showed examples like this one.
Please help me on the procedure and solution.

 

[micromass]

Hi hannahlu92!

Showing that an operator T is one-to-one is equivalent to showing that the kernel is 0. So, what you must show is that

$$T(a+bX+cX)=0$$

then a=b=c=0. Can you do that??

 

[Hurkyl]

I'm having difficulty because my teacher never showed examples like this one.
Please help me on the procedure and solution.

So what if the problem looks different? Why can't you solve the problem the way you would normally do?

(Actually, it's probably easier to solve these problems than others you have faced, since there are easier ways than using your knowledge of linear algebra)

 

[hannahlu92]

I don't know how to because I haven't done one before. My teacher writes definitions on the board and proofs, but no practical examples, so nothing is cementing in my brain.

 

[micromass]

I don't know how to because I haven't done one before. My teacher writes definitions on the board and proofs, but no practical examples, so nothing is cementing in my brain.

Certainly you don't need any practical examples to figure out when

$$T(a+bX+cX^2)=0$$

Just use the definition of T...

 

[Akater]

I don't know how to because I haven't done one before. My teacher writes definitions on the board and proofs, but no practical examples, so nothing is cementing in my brain.

In this case, proofs themselves are practical examples. That's the point of math courses.