Basis Definition and 1000 Threads

Hi Folks, I find this link http://mathworld.wolfram.com/VectorSpaceBasis.html confusing regarding linear independence. One of the requirement for a basis of a vector space is that the vectors in a set S are linearly independent and so this implies that the vector cannot be written in terms of...

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let M be an n-dimensional manifold and let (U,\phi) and (V,\psi) be two overlapping coordinate charts (i.e. U\cap V\neq\emptyset), with U,V\subset M, covering a neighbourhood of p\in M, such that...

Homework Statement Given that the primitive basis vectors of a lattice are ##\mathbf{a} = \frac{a}{2}(\mathbf{i}+\mathbf{j})##, ##\mathbf{b} = \frac{a}{2}(\mathbf{j}+\mathbf{k})##, ##\mathbf{c} = \frac{a}{2}(\mathbf{k}+\mathbf{i})##, where ##\mathbf{i}##, ##\mathbf{j}##, and ##\mathbf{k}## are...

Homework Statement Consider in the space ##\mathbb{R}^5## vectors ##\vec{v}_1 = (2,1, 1, 5, 3)^T## , ##\vec{v}_2 = (3, 2, 0, 0, 0)^T## , ##\vec{v}_3 = (1, 1, 50, 921, 0)^T##. a) Prove that these vectors are linearly independent. b) Complete this system of vectors to a basis. If you do part b)...

Hi! I'm struggling in identifying the lattice points and atom basis. As I understand in a cube, there are 8 lattice points, on on each corner of a cube. But in 2d it is any square between 4 points which are the lattice points. Is this correct? So if the points on the corners are the lattice...

Homework Statement Rewrite the state |ψ⟩ = √(1/2)(|0> + |1>) in the new basis. |3⟩ = √(1/3)|0⟩ + √(2/3)|1⟩ |4⟩ = √(2/3)|0⟩ − √(1/3)|1⟩ You may assume that |0⟩ and |1⟩ are orthonormal. Homework Equations The Attempt at a Solution [/B] I have a similar example in my notes however there...

I am a beginer. I have read that any given signal whether it simple or complex one,can be represented as summation of orthogonal basis functions.Here, what the terms orthogonal and basis functions denote in case of signals? Can anyone explain concept with an example?Also,what are the physical...

I know from the Fourier Analysis that any signal can be represented as summation of elementary signals i.e. basis functions .Likewise,any image can be represented as summation of Basis images. Is there any available code, or even an algorithm, that would allow me to compute Basis images of an...

I am attempting to answer the attached question. I have completed parts 1-4 and am struggling with part 5. 5. Prove that if a^{l_0}b_1^{l_1}...b_n^{l_n}=e then a^{l_0}=b_1^{l_1}=...=b_n^{l_n}=e If |a|>|b1|>|b2|>...>|bn| then I could raise both sides of a^{l_0}b_1^{l_1}...b_n^{l_n}=e to the...

Homework Statement I am given this state, which is the result of a lamba particle decaying into a proton and neutral pion. Initial j = 3/2. The final state can theoretically be written as: I have already determined that: alpha_p = Sqrt[2/3] beta_p = Sqrt[1/3] alpha_d = -/+ Sqrt[2/5]...

Homework Statement Here's my problem. I only need help with the bottom part, but if you could explain the problem more vividly that would help too. Homework Equations A = S-1BS (?) There aren't really any relevant equations. This part of linear algebra is getting really abstract, at least I...

Homework Statement Not a homework problem. Typically, we are given a matrix, then asked to find the basis for the kernel or image space of the matrix. I've never seen a problem that did the converse (i.e., given the matrix for the kernel/image space of some matrix, find some matrix). I was...

4b). How can I find a basis? I was thinking of the standard basis $\{1,x,x^2\}$, but that doesn't work under the scalar multiplication definition in the vector space. EDIT: I think it is $\{0,x,x^2\}$ and we take $1$ to be the $0$ vector! $a(0)+b(x)+c(x^2)=1$ implies $a=b=c=0$. It is strange...

What would be the effects on the system for different values of the Hamiltonian preferred basis in Decoherence? Would it for example make the electrons higher in orbital or bands? Or what would be the exact effects?

Suppose a change of basis from basis ##B## to basis ##C## is represented by the matrix ##S##. That is, ##S## is the transformation matrix from ##B## to ##C##. Now if ##t## is a given linear transformation, ##t:~V\rightarrow V##, with eigenvectors ##\epsilon_i##, say, and ##T## is the...

Homework Statement Verify that the functions y1(x) = x and y2(x) = 1/x are solutions of the differential equation y'' + (1/x)y' - (1/x2)y = 0 on I = (0,∞). Show that y1(x), y2(x) is a basis of the solution space of the differential equation. The Attempt at a Solution For the first part I'll...

Homework Statement The Matrix P = 1 0 3 1 1 0 0 3 1 is the transition matrix from what basis B to the basis B' = {(1,0,0),(1,1,0),(1,1,1) for R3? Homework Equations [v]B=P[v]B' The Attempt at a Solution I'm looking...

While I'm reading a book in quantum mechanics, I reached the part "Generalization to infinite dimension". We know that at infinite dimension many definitions changes.And that what is confusing me! Take for example the inner product.when we are dealing in finite dimension the definition of inner...

Find a basis for the solution space of the linear system x1-x2-2x3+x4 = 0 -3x1+3x2+x3-x4 = 0 2x1-2x2+x3 = 0 I created a matrix (not augmented, will be 0 on right side no matter what row operations) and brought it to reduced echelon form. x2 and x4 were free variables and I set them to the...

Homework Statement Suppose that ## u = s_1i + s_2j ## and ## v = t_1i + t_2j ##, where s1, s2, t1 and t2 are real numbers. Find a necessary and sufficient condition on these real numbers such that every vector in the plane of i and j can be expressed as a linear combination of the vectors u and...

Dear all, I have been searching some scientific basis about the most known -there are others less popular- complementary & alternative medicines listed below: - Aromatherapy - Ayurvedic Medicine - Bach Flowers - Chiropractic - Chromotherapy - Iridiology - Kinesiology - Oligotherapy -...

Please help me with these three questions. I'm really struggling to understand these concepts and I think that with an understanding of these three, I will be able to tackle the rest before my test on Wednesday. Thank you. http://www.texpaste.com/n/g4rwmzzw 1) $$ A = \left[\begin{matrix} -6 &...

Homework Statement Let V be a vector space, and suppose that \vec{v_1}, \vec{v_2}, ... \vec{v_n} is a basis of V. Let c\in\mathbb R be a scalar, and define \vec{w} = \vec{v_1} + c\vec{v_2}. Prove that \vec{w}, \vec{v_2}, ... , \vec{v_n} is also a basis of V. Homework Equations If two of the...

Hi, Can we treat prime numbers as an Ortho-normal basis of "Infinite" dimensions to represent every possible number. Treating numbers as vectors. Thanks.

Homework Statement Find a basis for and the dimension of the solution space of the homogenous system of equations. 2x1+2x2-x3+x5=0 -x1-x2+2x3-3x4+x5=0 x1+x2-2x3-x5=0 x3+x4+x5=0 Homework EquationsThe Attempt at a Solution I reduced the vector reduced row echelon form. However the second row...

Homework Statement si(t) = √(((2*E)/T)*cos(2*π*fc*t + i*(π/4))) for 0≤t≤T and 0 otherwise. Where i = 1, 2, 3, 4 and fc = nc/T, for some fixed integer nc. What is the dimensionality, N, of the space spanned by this set of signal? Find a set of orthonormal basis functions to represent this set of...

Hello I'm taking linear algebra and have a couple of questions about linear independence, spanning, and basis Let me start of by sharing what I think I understand. -If I have a matrix with several vectors and I reduce it to row echelon form and I get a pivot in every column then I can assume...

Let u = [1, 2, 3, -1, 2]T, v = [2, 4, 7, 2, -1]T in ℝ5. Find a basis of a space W such that w ⊥ u and w ⊥ v for all w ∈ W. I think the question is quite easy. Given this vector w in the space W is orthogonal to both u and v. I can only think of w being a zero vector. But would this be too...

Hi everyone, I'm trying to simulated the XRD pattern of Bi2Sr2Ca2Cu3O10, but I'm having a problem of finding the basis of atom(and their respective position). Also its JCPDS is quite hard to find, so if anyone working with this, may you provide a link or articles about my problem. Thanks...

Homework Statement How can I find the matrix representation of ##\mathbb{S}_+## and ##\mathbb{S}_-## in the ##|\pm y\rangle## or ##|\pm x\rangle## basis?Homework Equations ## \mathbb{\hat{S}}_+|s,m\rangle = \sqrt{s(s+1)-m(m+1)}\hbar|s,m+1\rangle ## The Attempt at a Solution The book almost...

Find a basis for $U=\text{span}{}\left\{\begin{bmatrix}1\\1\\0\\0\end{bmatrix}\begin{bmatrix}0\\0\\1\\1\end{bmatrix}\begin{bmatrix}1\\0\\1\\0\end{bmatrix}\begin{bmatrix}0\\1\\0\\1\end{bmatrix}\right\}$ Let $U=\text{col}(A)$, and applying row reduction on A, we obtain...

If we consider the singlet state (0,1,-1,0)/sqrt2 then it is easy to see that the unitary block transformation : A=RoR^-1with R a rotation of 45 degrees gives the vector 1/2(-1,1,-1,1) which is separable. Thus entanglement disappears in that basis.

How do we know that separable solutions of Schrodinger equation (in 3d) form a complete basis? I understand that the SE is a linear PDE and therefore every linear combination of the separable solutions will also be a solution , but how do we know that the converse, i.e 'every solution can be...

Dear All, Here is one of my doubts I encountered after studying many linear algebra books and texts. The Euclidean space is defined by introducing the so-called "standard" dot (or inner product) product in the form: (\boldsymbol{a},\boldsymbol{b}) = \sum \limits_{i} a_i b_i With that one...

Warning: this may be totally trivial, or totally wrong. I've been working through Sean Carroll's lecture notes, and I've got to http://preposterousuniverse.com/grnotes/grnotes-two.pdf . I follow the derivation for showing that the tangent space bases are the partial derivatives (Carroll's...

We are given the vectors la> = (1,0) and lb> = (0,1) and then a Hamiltonian H which is a 2x2 matrix with 2 on the diagonal entires and zero elsewhere. I am asked to now represent H in the basis of the vectors la'> = 1/sqrt(2)(1,1) and lb'> = 1/sqrt(2)(1,-1), which are also eigenvectors of H...

Hello all I am trying to solve this problem: Extend the following vectors to a basis of R^4. \[u_{1}=\left ( \begin{matrix} 1\\1 \\1 \\1 \end{matrix} \right )\] and \[u_{2}=\left ( \begin{matrix} 2\\2 \\3 \\4 \end{matrix} \right )\] What I did, I put these vectors as columns of a matrix...

what is the basis for all things that exist ? does it start with a single atom? are there atoms that exist in every thing in existence ? how are they different and how are they the same ?

Hello, I am just doing my homework and I believe that there is a fault in the problem set. Consider the set of functions defined by V= f : R → R such that f(x) = a + bx for some a, b ∈ R It is given that V is a vector space under the standard operations of pointwise addition and scalar...

Hello, I know that given a set $X$ and a topology $T$ on $X$ that a basis $B$ for $T$ is a collection of open sets of $T$ such that every open set of $T$ is the Union of sets in $B$. My question is: does taking the set of all Unions of sets in $B$ give exactly the topology $T$ ?

Homework Statement Let Amn be a matrix representation of some operator A in the basis |φn> and let Unj be a unitary operator that changes the basis |φn> to a new basis |ψj>. I am asked to write down the matrix representation of A in the new basis. Homework EquationsThe Attempt at a Solution...

Homework Statement Let S, a subspace of ℝ3 be the set of vectors orthogonal to vector (1,2,3) a)describe Set S b) find a basis for Set S 2. Relevant Equations That a basis has to be linearly independent and span R^3The Attempt at a Solution [/B] I would do this: I know that vector (1,2,3) is...

I'm just having a small trouble understanding the difference ( occurred while I was doing exercise). A basis is defined as 1)linearly independent 2)spans the space it is found in. Here is where I get confused: To determine whether or not a set spans a vector space, I was taught to find its...

The title may seem a little confusing and possibly stupid :D What I mean is like a plane doesn't go through the origin? Can we describe a basis for this? If so, how?

Homework Statement How do I find a basis for: the subspace of R^3 consisting of all vectors x such that x ⋅ (1,2,3) = 0. Homework Equations I believe this is performed through setting x = x,y,z, setting each parameter sequentially equal to 1 while the others are set to o, putting into a matrix...

I've attached the question to this post. The answer is true, but I'm trying to figure out why. Using Gram-Schmidt, I can only necessarily find 3 orthogonal vectors given 3 linearly independent vectors from ## R^5 ##. How then is it possible to extend this set of 3 vectors that are linearly...

Hello every one , in this pic i just printed ( Tensors_The Mathematics of Relativity Theory and Continuum Mechanics by Anadijiban Das ) here the author classifies the basis into 3 types 1- is the general basis (non-holomonic ) , 2- coordinate basis ( holomonic ),3- orthonormal basis ( non-...

Hi guys, I'm having a hard time with that one from Cohen-Tannoudji, ##F_{VI}## # 6. I'm translating from french so sorry if some sentence are weird or doesn't use the right words. 1. Homework Statement We consider a system of angular momentum l = 1; A basis from it sub-space of states is...

Say a subspace S of R^3 is spanned by a basis = <(-1,2,5),(3,0,3),(5,1,8)> By putting these vectors into a matrix and reducing it to rref, a basis for the row space can be found as <(1,-2,-5),(0,1,3)>. Furthermore, the book goes on to say that this basis spans the subspace S. Cool, not...

Homework Statement We have the initial orbital angular momentum state in the x basis as |l,ml>x=|1,1>x. We are asked to find the column vector in the z-basis that represents the initial orbital angular momentum of the above state. It then says "hint: use an eigenvalue equation". Homework...