Bases Definition and 294 Threads

Homework Statement Show that the set {sin(nx)} from n=1 to n=∞ is orthogonal bases for L^2(0, π). Homework EquationsThe Attempt at a Solution Proof: Let f(x)= sin(nx), consider scalar product in L^2(0, π) (ƒ_n , ƒ_m) = \int_{0}^π ƒ_n (x) ƒ_m (x) \, dx = \int_{0}^π sin(nx)sin(mx) \, dx =...

Homework Statement Which of the following would you not do in a laboratory setting? I. Pour acids and bases over a sink II. Wear goggles III. Heat a stoppered test tube (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III Homework Equations - The Attempt at a...

Homework Statement The pKa of acetylsalicylic acid (aspirin) is 3.5. The pH of gastric juice in the stomach is about 2-3 and the pH in the small intestine is about 8. Aspirin will be Answer is ionised in the small intestine and almost unionised in the stomach. Homework EquationsThe Attempt at...

Homework Statement 1) In a vector space V of all real polynomials of third degree or less find basis B such that for arbitrary polynomial p \in V the following applies: [p]_B = \begin{pmatrix} p'(0)\\p'(1)\\p(0)\\p(1)\end{pmatrix} where p' is the derivative of the polynomial p. Homework...

I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric compatibility in a non-coordinate basis, using the Ricci rotation coefficients...

Homework Statement Suppose a linear transformation T: [P][/2]→[R][/3] is defined by T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0) a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2]) b) Find the matrix representation of T (relative to standard...

Homework Statement Let B1={([u][/1]),([u][/2]),([u][/3])}={(1,1,1),(0,2,-1),(1,0,2)} and B2={([v][/1]),([v][/2]),([v][/3])}={(1,0,1),(1,-1,2),(0,2,1)} a) Show that B1 is a basis for [R][/3] b) Find the coordinates of w=(2,3,1) relative to B1 c)Given that B2 is a basis for [R[/3], find...

Hi All the chemical compounds can be categorized as acids, bases, and neutral compounds. There are different theories, namely Arrhenius Theory, Brønsted-Lowry Theory, Lewis Theory, which define the acids and bases. Each successive theory becoming more general; Lewis Theory is the most...

Hi everyone, I need some help to solve this problem: The direction states to find the value by using the log table $\log_{3}\left({825.6}\right)$ Work: I using the change of base: $\log_{3}\left({825.6}\right)=\frac{\log\left({825.6}\right)}{\log\left({3}\right)}$ I look up the values of...

Hi! I just read in an inorganic chemistry book (by Whitten, et al.) that acids are defined as substances that produce H+ ions in dilute aqueous solutions, and bases are those that produce OH-. To me, this definition implies that a substance that has yet to produce an H+ or an OH- can already be...

Ψ (x,0)= ΣcnΨn Ψ is a linear combination of all Ψn and Ψn are the bases. What does this statement actually mean..?? I mean, does it says that Ψ(x,0) contains c1 part of Ψ1 and c2 part of Ψ2 and so on...like we represent a vector : R= 3x+4y+7z...??

I can most of the time successfully convert between base 10 and another base or another base and base 10 or between 2 bases where one of them is a power of the other(like base 2 and base 4 or base 3 and base 9). With negative bases I sometimes don't get what I want in that negative base and...

Homework Statement Hi everyone. I'm a relatively new high school chemistry teacher. I'm teaching modified arrhenius theory to my students, and I'm unclear on ionization vs. dissociation. Homework EquationsThe Attempt at a Solution Here is my understanding: - Dissociation occurs when an ionic...

I've written the following brief history on how the definition of acids and bases change over time. Lavoisier came up with his definition circa 1776 in that 'p is acidic iff p contains much oxygen' and that held sway for 30 years[1] until it was falsified by Davy in 1810[2] who demonstrated...

Problem: Suppose u1,...un and v1,...vn are orthonormal bases for Rn. Construct the matrix A that transforms each vj into uj to give Av1=u1,...Avn=un. Answer key says A=UV^T since all σj=1. Why is all σj=1?

< Mentor Note -- thread moved to HH from the technical PF forums, so no HH Template is shown > Hello! So our teacher puts some really hard questions in acids and bases exams, and I would like someone to help me find similar questions and know how to solve these questions.. So the first...

After specifying the primitive vectors of a structure namely (a,b,c) in Wien2k, one has to specify locations of atoms in unit cell.There are three components X,Y,Z which determine the atom location. I want to know whether "X,Y,Z components" are corresponding to vectors a,b,c or they are...

Hi all, Just doing a bit of personal study on vector spaces and wanted to clear up my understanding on the following. This is my description of what I'm trying to understand, is it along the right lines? (apologies in advance, I am a physicist, not a pure mathematician, so there are most...

Hi All struggling with concepts involved here So I have {P}_{2} = \left\{ a{t}^{2}+bt+c \mid a,b,c\epsilon R\right\} is a real vector space with respect to the usual addition of polynomials and multiplication of a polynomial by a constant. I need to show that both...

Please look at the image. For (b) and (c), what the solution manual says makes no sense to me. You're reacting a primary alkyl halide with an unhindered base EtONa. They say that the only product is the alkene. Why do they not consider the SN2 reaction that WILL take place in larger quantity...

To represent operations of a point group by matrix we need to choose basis for this representation. What is the criteria for doing that? How to realize that how many bases are necessary for a matrix representation and how to select them? Or could you please give me an elementary reference to...

Homework Statement Let So = {v1,v2,v3,v4} be a basis of the vector space V. S= {u1,u2,u3,u4} is a set of vectors defined as follows: u1 = 80v1 + 106v2 + 120v3 +164v4 u2 = 80v1 + 146v2 + 136v3 + 91v4 u3 = 90v1 + 143v2 + 122v3 + 70v4 u4 = 80v1 + 56v2 + 80v3 + 48v4 Find the Transition...

Question statement: We are given that Ca and CaF2 are both Ca face-centred cubic lattices, and that in the case of CaF2 there is a basis of F ions at +/-(1/4, 1/4, 1/4). Then explain qualitatively how the ratio of the (2 0 0) and (4 0 0) Bragg peak intensities in the X-ray diffraction patterns...

Homework Statement Let So = {v1,v2,v3,v4} be basis of vector space V. And S = {u1,u2,u3,u4} be set of vectors defined as follows: u1 = 20v1 + 46v2 + 116v3 + 170v4 u2 = 20v1 + 86v2 + 147v3 + 174v4 u3 = 30v1 + 89v2 + 59v3 + 81v4 u4 = 15v1 + 27v2 + 12v3 + 9v4 Find transition matrix A from...

Homework Statement Let V be the subspace of R3 defined by V = {(x,y,z) | x - y +2z = 0} Then A = {(2,0,-1) , (1,1,0)} B = {(1,3,1) , (3,1,-1)} are both bases for V. Show. The Attempt at a Solution 1) check that both set A and set B of vectors are in R3. V = (x , y ...

http://i.imgur.com/ORtBJdT.jpg i don't understand why the old base is written in terms/as a linear combination of the new bases. wouldn't i want to map my coordinates from old to new not new to old?.. here's what my textbook says about it, can you guys interpret this for me, i still don't...

Homework Statement convert 14.334 to the base 16 to a no. to the base 10 Homework Equations sum ofeach no.*its poand take position weigth The Attempt at a Solution the integer part 14 will be 4*16^0+1*16^1=20 to the base 10 the decimal part 0.334 is my problem i multiply 0.334 *16...

Imagine there is a reaction performed and it contains a product that is easily separable via an acid base extraction. This product is only formed in 50% yield. The other 50% formed unwanted products that are to be separated and done away with. If all that is known is the structure of these...

Homework Statement Let ##S = \{1, e^x, e^{-x}, e^{2x}, e^{-2x}\}## and ##B = \{1, sinh(x),cosh(x), sinh(2x), cosh(2x)\}##. S spans the vector space V, and a linear transformation T: V -> V is defined by T(y) = y'' - 3y' - 4y. (a) Find the representation matrix of T with respect to the bases S...

Homework Statement what is the H, Oh , ph and poh for a 0.048 mol/l solution of benzoic acid? C6H5COOH Homework Equations Ph +poh=14 Ph=-log(h30+) H30=10^-ph The Attempt at a Solution I thought maybe u need to separate benzoic acid into its ions and then use the molar ratio to...

Homework Statement The vector space V is equipped with a hermitian scalar product and an orthonormal basis e1, ..., en. A second orthonormal basis, e1', ..., en' is related to the first one by \mathbf{e}_j^{'}= \displaystyle\sum_{i=1}^n U_{ij}\mathbf{e}_i where Uij are complex numbers...

I'm trying to solve a dynamical quantum mechanics problem related to the Cs atom, but I'm having trouble in the following, and I'm afraid I'm doing it wrong. Say I have the matrix form of the Hamiltonian on a basis for a system | \psi \rangle to be H_\psi, and another system with bases |...

Homework Statement What is the H3O+ concentration of a 0.100 M acetic acid solution (Ka = 1.8 x 10-5)? a. 1.8 x 10-5 b. 1.8 x 10-4 c. 1.3 x 10-2 d. 1.3 x 10-3 e. 0.9 x 10-3 Homework Equations The Attempt at a Solution so the question is supposed to be D. but I thought that...

Homework Statement If a weak acid, HA, is 3% dissociated in a 0.25 M solution, calculate the Ka and the pH of the solution. Homework Equations pH = (pKa - log[HA])/2 The Attempt at a Solution Honestly, I'm not even sure where to start. I don't have the pH or the pKa - how am I...

Hello PF members! I have a problem regarding coordinate and non-coordinate bases. As I understood from my course in GR, the partial derivatives of a coordiante system always commute: [\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}] = 0 . Which is not necessarly true for...

Here is the question: Here is a link to the question: Let {e1, e2, e3} be a basis for the vector space V and T: V -> V a linear transformation.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

On the Wikipedia page for Radix Economy it shows how to find the economy (efficiency for data storage) of a positive real base like 2, e, 3, pi, etc but how would I find the economy of a complex or negative base? http://en.wikipedia.org/wiki/Radix_economy I want to find the economy of Donald...

Please I have read texts about changing the bases of numbers but still I have difficulties in doing, may s'body instruct me for example.. Change 3145 to base 8 Thanx..

I am reading Dummit and Foote Section 9.6 Polynomials in Several Variables Over a Field and Grobner Bases. I have a second problem (see previous post for first problem) understanding a step in the proof of Proposition 24, Page 322 of D&F Proposition 24 reads as follows: Proposition 24. Fix a...

Lets say a certain chemical reacts with an acid. Does that make that chemical a base always and vice versa? Like in S + NaOH → Na2S + Na2S2O3, Sulfur reacting with a base would it make S an acid? Or do we call it a base or an acid, reacting with the opposite if it's producing water as a...

Two questions: If you have two states which have at least one common eigenvalue, then are the two states distinguishable? If you have one state but measure it with two different bases, can one conclude anything if the two measurements have a common eigenvalue? Thanks

In ordinary QFT, everything is formulated in terms of a Fock basis so when we write |\psi\rangle we mean that this is a product of single particle states covering every momentum mode. This leads to a Hamiltonian that's typically of the form \hat H=\int \frac{d^{3}k}{(2\pi)^{3}} [\omega_{k}(\hat...

Homework Statement Let B be an ordered orthonormal basis for a k-dimensional subspace V of ℝn. Prove that for all v1,v2 ∈ V, v1·v2 = [v1]B · [v2]B, where the first dot product takes place in ℝn and the second takes place in ℝk. Homework Equations The Attempt at a Solution Let B...

Let B and C be ordered bases for ℝn. Let P be the matrix whose columns are the vectors in B and let Q be the matrix whose columns are the vectors in C. Prove that the transition matrix from B to C equals Q-1P. I am stuck. Here is what I have. I know that if B is the standard basis in ℝn...

Homework Statement Let B and C be ordered bases for ℝn. Let P be the matrix whose columns are the vectors in B and let Q be the matrix whose columns are the vectors in C. Prove that the transition matrix from B to C equals Q-1P. Homework Equations An ordered basis for a vector space V is...

Write down orthonormal bases for the four fundamental subspaces [...]" Homework Statement Problem: Write down orthonormal bases for the four fundamental subspaces of A = matrix([1,2],[3,6]]). (1 and 2 are on the first row whereas 3 and 6 are on the second row.) Solution: A =...

Homework Statement Determine whether each of the following is a subspace of ℝ3, and if so find a basis for it: a) The set of all vectors (x, y) b) The set of all vectors of the form (sin2t, sintcost, 3sin2t) Can someone please explain to me how you determine whether these are...

Homework Statement solve the equation log4 x = 1 + log 2 2x , x>0 My attempt I changed it base 2 so I now have( when i write log it's to the base 2) log2x/( log2 4) = log 2 + log 2x and I'm stuck here I keep trying to form an equation, but I can't.

Could we have close-packed structures with bases? Or, is it possible in close-packed structures that atoms of bases have arbitrary sizes? For example could we have a fcc close-packed structure with a two-atomic base (of any types of atoms)? If yes, what would happen if sizes of the bases be...

Homework Statement Oxalic acid, H2C2O4, which is found in certain pants, can provide two hydronium ions in water. Write balanced equations to show how oxalic acid can supply one and then a second H3O+ ion. Homework Equations There's not really an equation, and if there is please don't make...