Basis for P2 and Linear Transformation

[Shay10825]

Hello. I'm having some trouble with this problem. Any help would be greatly appreciated.

Homework Statement

Consider B= (2x+3, 3x^2 +1, -5x^2 + x-1}

a) Prove that B is a basis for P_2

b) Express -x^2 - 2 as a linear combination of the elements of B

c) If t: P_2 -> P_2 is a linear transformation, and T(2x+3) = x^2 -1, T(3x^2 +1) = x^2 -2x, T(-5x^2 +x-1)= -x^2 +3x, compute T(-x^2 -2) 2. The attempt at a solution

a)
Since:
B spans P2 and B is linearly independent, B is a basis for P2

b)
B = {2x+3, 3x2+1, 5x2+x-1}
2x t 3 = a
3x2 +1 = b
-5x2 + x -1 = c

By inspection:
-x2 - 2 = 2c + 3b –a
-x2 - 2 = 2(-5x2 + x -1) + 3(3x2 +1) – (2x t 3)

c) I typed it in Microsoft Word and uploaded it here:

http://img340.imageshack.us/img340/6180/matrixtheorydj6.png

Thanks

 

[lalbatros]

For point a), I guess you need to prove it.
At least you need to prove the linear independence.
How can you prove that these three polynoms (vectors) are linearly independent?
Check your textbook.
Counting and using a theorem should then end the proof. Textbook again.

For point b), you don't show your derivation and a solution in itself is not interresting.
Moreover it is probably wrong, if I can guess correctly what your wrote.
Assuming you wanted to define the following notations (check for some mistakes):

a = 2 x + 3
b = 3 x2 + 1
c = 5 x2 + x - 1

then simply checking your proposal gives:

2c + 3b - a = -2 + 19 x2 and this is different from -x^2 - 2

Therefore, as far as I guessed correctly from your post, your answer is wrong.

For point c), this is straightforward if you have solved point b) correctly.
This is just a substitution and a few calculations.
Remember what a linear transformation is (see textbook).
What are the main properties defining a linear transformation?

 

[Shay10825]

For point b), you don't show your derivation and a solution in itself is not interresting.
Moreover it is probably wrong, if I can guess correctly what your wrote.
Assuming you wanted to define the following notations (check for some mistakes):

a = 2 x + 3
b = 3 x2 + 1
c = 5 x2 + x - 1

then simply checking your proposal gives:

2c + 3b - a = -2 + 19 x2 and this is different from -x^2 - 2

Therefore, as far as I guessed correctly from your post, your answer is wrong.

For point c), this is straightforward if you have solved point b) correctly.
This is just a substitution and a few calculations.
Remember what a linear transformation is (see textbook).
What are the main properties defining a linear transformation?

b) it was suppose to be
a = 2 x + 3
b = 3 x2 + 1
c = -5 x2 + x - 1

 

[lalbatros]

Then I guess the linear combination you gave is correct.
For point c, you simply need to reproduce the same combination on the tansfromed vectors to get transformed of the combination.
You should show clearly how you did to solve point b). Should be a system of equations...
For point a), prove the independence.