Linear Algebra - Representing Matrix


by Mumba
Tags: algebra, linear, matrix, representing
Mumba
Mumba is offline
#1
Mar31-10, 12:18 PM
P: 27
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
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Mark44
Mark44 is offline
#2
Mar31-10, 12:28 PM
Mentor
P: 20,937
Quote Quote by Mumba View Post
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
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
Mumba is offline
#3
Mar31-10, 12:32 PM
P: 27
yes, we called it R.
sorry i meant the same. A transformations from <=2 to <=3....

Mark44
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#4
Mar31-10, 01:00 PM
Mentor
P: 20,937

Linear Algebra - Representing Matrix


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
Mumba is offline
#5
Mar31-10, 01:10 PM
P: 27
Quote Quote by Mark44 View Post
For this problem I would advise you to evaluate L at each of the three basis polynomials.
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
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#6
Mar31-10, 02:20 PM
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P: 20,937
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
Mumba is offline
#7
Mar31-10, 02:33 PM
P: 27
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
Mumba is offline
#8
Mar31-10, 03:53 PM
P: 27
yeah its good
thx mate!!! ;)


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