Recent content by maximus101

Homework Statement For x > 0, define y(x) := x \int_0^{log x} \! \sqrt{1 + e^t} dt -(2/3)(1 + x)^{3/2} Calculate y'(x) := dy/dx and y''(x) := d^2y/dx^2 Homework Equations The Attempt at a Solution would like to work through this with person/people please. Not sure...

great got it, thanks

if this is correct, could you possibly explain how Rank (A) > Rank (B) where B is the m x n matrix we get from A by removing the last column .

ok I get it now, we got the system of equations and we showed that (ii) holds is the same as saying there is no solutions of the linear equations which is exactly what one says (i) ?

okay, in this case, why is dim U > Rank T ? I'm not sure how to use the fact that it is non-singular

okay thanks, that is a lot more clear now we will have m equations: \alpha_1c_{1,1}+...+\alpha_nc_{1,n}=c_{1,(n+1)} . . . . \alpha_1c_{m,1}+...+\alpha_nc_{m,n}=c_{m,(n+1)} we assumed linear dependence, so I think we are trying to prove by counterexample that linear...

okay I think I understand a bit better, so considering c1 this will form the first colum of A and it will have m rows, so we will have these equations, for the first row of A it will be c11c1 + c21c2 + ... + cn1cn = c(n+1)1 were cij s.t. i=column j=row and the second...

Hey, thank you I understand now. Could you give me an example where (dimU + dimV)/2 > rankT

Okay I think perhaps from this information I could get a_1c_1 = c_{n+1} - a_2c_2 - a_3c_3 - ... - a_nc_n and then do this for c_2 ... c_n and have n equations but I'm not sure how we would get a system of m equations from this, unles we made a matrix with m rows. Also I see the...

Homework Statement If we let A be the augmented m x (n + 1) matrix of a system of m linear equations with n unknowns Let B be the m x n matrix obtained from A by removing the last column. Let C be the matrix in row reduced form obtained from A by elementary row operations...

Is this statement true or false if false a counterexample is needed if true then an explanation If T : U \rightarrow V is a linear map, then Rank(T) \leq (dim(U) + dim(V ))/2

For a fixed y \in R , if f(x) = [sin(x+y)]-(sin x) (cos y) − (cos x) (sin y)and we let E(x) = [f(x)]^2+[f'(x)]^2. How do we prove the addition formulae for sin(x + y)and cos(x + y) by applying the Mean Value Theorem to E

so I did the following: \dfrac{|f(y) - f(x)|}{|y-x|}\leqslant |y-x| was obtained by (y-x)^2=|y-x||y-x| and dividing both sides by |y-x|. then trying to use the fact that \displaystyle\lim_{y\to x}\frac{f(y)-f(x)}{y-x} is the derivative at x but not sure what's next how to...

sorry just fixed it

for which f: R \rightarrow R such that \forall x,y\in R does | f(y) - f(x) | \mid \leq (y-x)^2 hold