Need some Linear Algebra help

[hans030390]

I need some help on this Linear Algebra problem, if not just the steps to work through it. Sorry that the formatting isn't perfect.

Let V = P2 be the space of polynomials of degree at most 2, and W = R^3.
Let L : V --> W be given by L(f) = (f(1), f'(-1), f''(0)). With respect to
the basis f1 = 1, f2 = x, f3 = x^2 of V and the standard basis e1 = (1, 0, 0),
e2 = (0, 1, 0), e3 = (0, 0, 1) of R^3 find the matrix A representing L.

The solution he gave the class is extremely vague, and we haven't worked on a problem too similar to this before. Any help is really appreciated. I may have more questions later.

 

[hans030390]

I also need help on this one, if possible. Again, thanks a lot of you can help out.

Let V = R^4 with the inner product <v, w> = v - w (dot in between instead of dash), and let W be the 2D subspace spanned by the vectors e1 = (0, 1, 2, -2) and e2 = (-6, 4, 8, 1). Find the orthogonal projection of the vector (5, 8, -2, -1) onto W.

Steps are to apply the Gram-Schmidt process to e1, e2 to obtain f1 and f2. I was able to get f1 as 1/3(0, 1, 2, -2), and f2 is supposed to be 1/9(-6, 2, 4, 5). The projection is then (2, 0, 0, -3) by (v (dot) f1)f1 + (v (dot) f2)f2.

Like I said, I was able to get f1, but am not really sure how to get f2...so, obviously, I'm unsure how to do the rest either.

 

[dighn]

basically you want a matrix A, where W = A*V, and where V are formed with components in the x^n basis i.e. the coefficients in a + b*x + c*x^2 form a vector (a, b, c), and W is just the vector of reals as shown
so W is just (a + b + c, b - 2c, 0), and W = A * V putting A in matrix form you get ( (1,1,1), (0, 1, -2), (0,0,0) )

 

[hans030390]

Originally posted by: dighn
basically you want a matrix A, where W = A*V, and where V are formed with components in the x^n basis i.e. the coefficients in a + b*x + c*x^2 form a vector (a, b, c), and W is just the vector of reals as shown
so W is just (a + b + c, b - 2c, 0), and W = A * V putting A in matrix form you get ( (1,1,1), (0, 1, -2), (0,0,0) )

*Sigh*

I obviously haven't learned a single thing in this class, because most of that just went over my head (my teacher is horrible, and I'm not one to usually blame the teacher). That, and I have solutions right in front of me, and I'm not seeing any of that...

Yeah, I know, I have the answers, but the steps are horribly vague. Guess that's what I get for missing the review, huh?

 

[dighn]

well first of all you need to understand the concept of basis properly. (actually I hope my answer is correct?)
if you have a polynomial V = f(x) = a + bx + cx^2 and if you can represent it with a vector (v1, v2, v3), and if you choose your basis to be f1 = 1, f2= x, and f3=x^2, all that means is that V = v1 * f1 + v2 * f2 + v3 * f3 = a + bx + cx^2. from that you get v1 = a, v2 = b, v3 = c. so representation of V in the given basis is (a, b, c), for a + bx + cx^2.
then you have W = (f(1), f'(-1), f"(0)) which is given in the basis the question wants it in. you can verify this by making a basis representation (w1, w2, w3), which means that w1 * (1 0 0) + w2 * (0 1 0) + w3*(0 0 1) = (f(1), f'(-1), f"(0)) = (w1, w2, w3), so w1 = f(1), w2 = f'(-1), w3 = f"(...), and the representation is still (f(1), f'(-1), f"(0)).
then you evaluate (f(1), f'(-1), f"(0)) and that's W
once you have W and L in W = A*L, it's just a matter of lining up the a b c's and extracting the matrix elements.