Recent content by damabo

Homework Statement Given are two relations for the molar volume. Are they possible? If so, give the formula for v in function of P and T. a) dv =R/P dT - RT/P² dP b) dv = 2R/P dT - RT/2P² dP Homework Equations The Attempt at a Solution If I integrate dv I get ∫R/P dT - ∫RT/P² dP= RT/P +...

well, first you prove that v_1 and v_2 are orthonormal: normal: sqrt(cos²(theta) + sin²(theta))=1 in both cases. ortho: if [.,.] is the inproduct in R, then [v_1,v_2]= (-sin(th)cos(th)+sin(th)cos(th))=0. then you prove that they are a basis of R^2. which means 1. linearly independent 2...

we must have A.u_1=v_1=(cos(theta),sin(theta))^T , and A.u_2=v_2=(-sin(theta),cos(theta)) In other words, a_11 + a_12 = cos(theta) ; a_21 + a_22 = sin(theta) ; - a_12 = -sin(theta) ; - a_22 = cos(theta) .

do you want to evaluate \frac{x^7}{2} or (x/2)^7 ? in the first case, we get (\frac{(x+h)^7}{2} - \frac{x^7}{2})/h . evaluating (x+h)^7 would require the binomial theorem, which says that (x+y)^n = Ʃfrom k to n \frac{n!}{k!(n-k)!}x^{n-k}.y^k

you better use \frac{a}{b} to make clear what fraction you want to make. make sure to use [itex][/itex ] (but without the latter space before the bracket) to place the expression you want between those two.

the definition of an isomorphism is that the function is bijective and linear. I think the first question pertains to the identical function, which does not (?) make use of * .

surjective and injective ofcourse: injective: I must show that if X_1 != X_2 then A.X_1 ≠A.X_2. So choose X_1,X_2 \in ℝ^n. because [AX_1,AX_2]=X_1^T.A^T.A.X_2 = X_1^T.X_2=[X_1,X_2] , length, distance and orthogonality will be preserved. so A.X_1 ≠A.X_2 . surjective: I must show that for...

Homework Statement if L_A: ℝ^n -> ℝ^n : X-> A.X is a linear transformation, and A is an orthogonal matrix, show that L_A is an isomorphism. also given is that (ℝ,ℝ^n,+,[.,.]) , the standard Euclidian space which has inproduct [X,Y]= X^T.Y Homework Equations ortogonal matrix, so A^T=A^{-1}...

(x+y)²=x²+y²+2xy . we know that x²+y²= ∞ However, this means that there are many possibilities: x = ± ∞ or y=±∞. there are the following possibilities: x=+∞ and y=+∞ ; x=+∞ and y=-∞ ; x= + ∞ and y is a finite number greater than 0 ; x = + ∞ and y is a finite number smaller than 0 ; x=-∞ a and...

see the problem statement: U\bot is indeed the orthogonal complement and V is the Euclidian space of which U is a subspace. U\bot\bot is thus the orthogonal complement of the orthogonla complement.

Homework Statement show that (U\bot)\bot=U, if (ℝ,V,+,[,.,]) an Euclidian space en U is a linear subspace of V. Homework Equations The Attempt at a Solution suppose \beta={u_1,...,u_k} is an orthonormal basis of U. pick u in U. Then u=x_1u_1+...+x_ku_k for certain x_1,...,x_k...

so everybody thinks that mathematicians (whether specialized in analysis, geometry, or algebra) can choose topics in physics - I mean professors, but also post-doc researchers and people working on their PhD ?

So, first, we need n independent eigenvectors to form a basis for V. this means that the sum of d_i's will be n. I understand there is a theorem which says that d_i ≤ m_i, because phi will be of the form ( X - λ )^d * p(X). This means that there could be more zero-points of phi, than d. So...

Homework Statement V is a linear vector space of dimension n \phi = det(X I_n -A) equals a product of first degree factors. Spec(L)={λ_1,...,λ_k} is the set of eigenvalues show that: if L is diagonizable than d(λ_i)=m(λ_i) Homework Equations d_i=d(λ_i)= geometric multiplicity = dim E_i =...

thanks, will apply that formula.