Linear Algebra - Representing Matrix

Mumba

1. The problem statement, all variables and given/known data
The Question:

The map is given: [tex]L\rightarrow \Re_{2} \rightarrow \Re_{3}, p \rightarrow[/tex] p' + q*p , with q(x) = x.
Now i should find the representing matrix for L with respect to the bases {1+x, x+x², 1+x²} for [tex] \Re_{2}[/tex] and {1,x,x+x²,1+x³} for [tex] \Re_{3}[/tex].


3. The attempt at a solution

I dont know what i should do here ^^. But i tried to calculate it.
p is the polynomial, so i thought its like this:
p = a₀(1+x) + a₂(x+x^2) + a₃(1+x^2)

So i can calculate [tex]p\rightarrow p' +q*p:[/tex]
[tex]p\rightarrow a_{0}+a_{1} + 3a_{1}x + a_{0}x + 2a_{2}x^2 + 2a_{1}x^2 + 2a_{2}x^3[/tex]

Then i just counted the coressponding values together, means
a₀ ~ 1+x --> 1 + 1 + 0 + 0
a₁ ~ x+x² --> 1 + 3 + 2 + 0
a₂ ~ 1+x² --> 0 + 0 + 2 + 2

Puting this all together gives the repr. Matrix
| 1 1 0 |
| 1 3 0 |
| 0 2 2 |
| 0 0 2 |

Is this somehow correct or completely wrong? ^^
I'm missing the second base here..
Does anyone knows a good website (linear algebra 1) ? I have exam in little more than week and still a lot to learn.

Thx
Mumba

Mark44

Your transformation is not a map from R² to R³ - it's a map from P₂ to P₃, where P₂ is the function space of polynomials of degree <= 2, and P₃ is the function space of polynomials of degree <= 3.

Mumba

yes, we called it R.
sorry i meant the same. A transformations from <=2 to <=3....

Mark44

But R² is not the same as P₂, nor is R³ the same as P₃. The dimension of R² is 2, while the dimension of P₂ is 3.

For this problem I would advise you to evaluate L at each of the three basis polynomials. Then write those three output polynomials in terms of the second basis. That should give you an idea of what the matrix for this transformation is.

Mumba

But this is what i wanted to do ^^.
But i dont know how. I ve never seen this before...
Thats why i asked is that Polynomial correct the way i have written it down?

Maybe you can give me an example, lets say for 1+x...
What should i do with this?
Sorry but i really dont know...:(

Mark44

Your formula is L(p) = p' + xp, so L(1 + x) = 1 + x(1 + x) = 1 + x + x^2. Do the same thing for the other two basis polynomials.

Mumba

Thx!!!!!!!!!!!!!!!!!!
then i get for:
L(x+x^2)=1+2x+x(x+^2)= 1 +2x+x^2+x^3
L(1+x^2)=2x+x(1+x^2)=3x+x^3

You said: write those three output polynomials in terms of the second basis.

so if i did this correct i get for
1+x --> 1 0 1 0
x+x^2 --> 0 1 1 1
1+x^2 --> -1 3 0 1

so my matrix is
1 0 -1
0 1 3
1 1 0
0 1 1

is this ok?

Mumba

yeah its good
thx mate!!! ;)