Matrix in bases P2 and R2

[Hot_Biscotti1274]

Hey! I am having some trouble solvning this question on an old exam.

" The linear map F : P2 →R2 is defined by:
F (p(x)) =(p(0), p(1))

a) Find the matrix of F with respect to the standard bases in P2 and R2. (The standard
basis in P2 is (1, x, x2).) "

What is (p(0), p(1))? How do I solve this question? I've seen the answer but can't wrap my head around it.

 

[Dr.Peterson]

Hey! I am having some trouble solvning this question on an old exam.

" The linear map F : P2 →R2 is defined by:
F (p(x)) =(p(0), p(1))

a) Find the matrix of F with respect to the standard bases in P2 and R2. (The standard
basis in P2 is (1, x, x2).) "

What is (p(0), p(1))? How do I solve this question? I've seen the answer but can't wrap my head around it.

I find examples helpful for understanding the meaning of something abstract.

Take the polynomial p(x) = 3x^2-4x+5 in P₂. Then the map F takes p to the ordered pair (p(0), p(1)) = (5, 4) in R². since p(0) = 3(0)^2-4(0)+5 = 5, and p(1) = 3(1)^2-4(1)+5 = 4. Does that make sense?

Then what are the images of the three basis polynomials?

 

[Steven G]

To continue with what Dr Peterson was saying.
F : P² →R² means that F takes objects from P² to R²
What exactly is P²? It is the set of ALL polynomials of degree 2 or lower.
What exactly is R²? It is set where each entry is in the form of (x,y) where x and y are any real numbers.
Now F(0) is just a real number. It is just function evaluation. Likewise F(1) is just a real number. So in the end (F(0),F(1)) is of the form (x,y) and lives in R².

What would (1, x, x²) equal if x=0. How about when x=1?

Please do not write x2 for x². You can write x² as x^2

 

[Hot_Biscotti1274]

Yes I understand that I have to put in p(x)=1, p(x)=x and p(x)=x^2 but I don't understand in what equation I should put it in? How do I put in those values to get a vector in R^2?

***I understood the question now and solved it. Thanks for your help!

 

[Steven G]

Yes I understand that I have to put in p(x)=1, p(x)=x and p(x)=x^2 but I don't understand in what equation I should put it in? How do I put in those values to get a vector in R^2?

No, you are mistaken. F takes one p(x) at a time and maps it to (p(0),p(1)). What you wrote above clearly has multiple p(x)'s!
p(x) can not equal 1, x and x^2!!!

Let p(x) = a + bx + cx^2 be the general degree 2 polynomial. In fact, if a=0, then p(x) would be a 1st degree polynomial, b is not 0, If a=b=0....
Now p(0) =a and p(1) = a+b+c

So F(p(x)) = (a, a+b +c)^T
Now you need to find a matrix A, such that Ap(x) = (a, a+b +c)^T