2 linear operator problems

1. Prove that mapping

$\displaystyle T:\mathbb{R}_{2}[x]\rightarrow \mathbb{R}_{2}[x] , T(ax^{2}+bx+c)= 3ax^{2}+2(b-c)x +a $

is linear operator and find some basis of subspace Im(T), and Ker (T),and find dim Im(T) and dim Ker(T)

It's easy to prove that is linear operator. But, how to find basis for Im, and Ker ?

2. Let A $\displaystyle A:\mathbb{R}^{3}\rightarrow R_{3}[x]$ be linear operator such that

$\displaystyle A(1,0,0)=2x + x^{3}, A(0,1,0)=-2x + x^{2}, A(0,0,1)= x^{2}+x^{3}$. Find linear operator a, some basis of subspace Im(T), and Ker (T),and find dim Im(T) and dim Ker(T).

In this one I dont know how to find A, and later, basis of Im and Ker.

Re: 2 linear operator problems

Quote:

Originally Posted by

**DonnieDarko** 1. Prove that mapping

$\displaystyle T:\mathbb{R}_{2}[x]\rightarrow \mathbb{R}_{2}[x] , T(ax^{2}+bx+c)= 3ax^{2}+2(b-c)x +a $

is linear operator and find some basis of subspace Im(T), and Ker (T),and find dim Im(T) and dim Ker(T)

It's easy to prove that is linear operator. But, how to find basis for Im, and Ker ?

Find three concrete polynomials that are mapped to zero. Does it tell you the general form of such polynomials? This is Ker(T). Once you figure out the general form of elements of Ker(T), finding the basis and dim Ker(T) should be easy. Also, dim Im(T) = dim $\displaystyle \mathbb{R}_{2}[x]$ - dim Ker(T).

Quote:

Originally Posted by

**DonnieDarko** 2. Let A $\displaystyle A:\mathbb{R}^{3}\rightarrow R_{3}[x]$ be linear operator such that

$\displaystyle A(1,0,0)=2x + x^{3}, A(0,1,0)=-2x + x^{2}, A(0,0,1)= x^{2}+x^{3}$. Find linear operator a, some basis of subspace Im(T), and Ker (T),and find dim Im(T) and dim Ker(T).

In this one I dont know how to find A, and later, basis of Im and Ker.

Can't you write A(a, b, c) using linearity of A?