Recent content by simpledude

Anyone?

Homework Statement Construct an orthogonal basis of R2 equipped with the non-standard inner product defined for all X, Y belonging to R2, by <X,Y> = X^T AY with A = 2 1 1 3 The Attempt at a Solution So it seems pretty trivial, but I can't seem to get the answer. So my approach is 1)...

Hi Dan, thanks for the last post. So I worked on the problem this morning, and when I get to the part F(G(a+bx)) = a+bx Assuming G(a+bx)=c+dx We can write this as F(c+dx) = a + bx So now is where I get lost, since if I solve for cx + d = a + bx I just get c = b and d=a ...

Yes did exactly that, by the way last equation should be -b-c So after this I solve the matrix of coefficients to see how many independent columns I have: 2 2 2 0 -1 1 0 2 -1 0 1 2 0 -1 -1 0 and simplified matrix is: 1 1 1 0 0 1 2 2 0 0 1 2 0 0 0 4 I get 4...

EDIT: Oh wait found a mistake in my math

I get dimension of Ker = 4, What I did is write out the matrix and multiply it out (since we know A and M I took as a,b,c,d). After multiplying and adding, I get a system of 4 equations, (a,b,c) and solve them via Gauss to find how many are independent. Is this ok?

Homework Statement Let V = M2(R) be the vector space over R of 2×2 real matrices. We consider the mapping F : V −> V defined for all matrix M belonging to V , by F(M) = AM +MA^T where A^T denotes the transpose matrix of the matrix A given below A =  1 2 −1 0  Question is...

Yeah, I just cleared it up, it is degree < 2. The approach is the same, so thanks Halls/Dan. I am having some issues with another part of the question, which asks to find an inverse mapping F^-1 I understand that since F is nonsingular we can find F^-1 : V-->V Furthermore, since the dimensions...

Ok so I got a0 = a1 = a2 = 0 (I used that instead of a,b,c) So can I say, since the only way to obtain {0} with this equation is by a0=a1=a2=0, we see that F is indeed nonsingular?

Ah! Thanks Dan, working it out now :)

Yes I did mean degree, sorry that's how we refer to it in our class :) Why can't I just solve P'(x) + P(x) = 0 for p(x) explicitly? Then get something of the form p(x) = A e^-x ?

Homework Statement Let V = P2(R) be the vector space of all polynomials P : R −> R that have order less than 2. We consider the mapping F : V −> V defined for all P belonging to V , by F(P(x)) = P'(x)+P(x) where P'(x) denotes the first derivative of the polynomial P. Question is: Show...

Thank you sir!

hehe thanks! However, is my approach correct? Can infinity be used? Also Dan, if you don't mind me asking a more abstract question. Let's say I wanted to show that T was a linear operator. Can I independantly show each part of the operator? I mean can I show u''(x) is linear, then...

Ok so more abstratly, Let's take an arbitrarily large x, x --> infinity The equation e^x will go to infinity The equation e^-2x will go to zero. So a(infinity) + b(0) = 0 is only satisfied if a=0 Now let's take x --> -(infinity) Th equation e^x will go to zero. The equation...