Basis Definition and 1000 Threads

I am reading Segei Winitzki's book: Linear Algebra via Exterior Products ... I am currently focused on Section 1.6: Dual (conjugate) vector space ... ... I need help in order to get a clear understanding of an aspect of the notion or concept of the dual basis \{ e^*_1, e^*_2, \ ... \ ... \...

Consider the following set of vectors in $\mathbb{R}^3:$ $u_0 = (1,2,0),~ u_1 = (1,2,1), ~u_2 = (2,3,0), ~u_3 = (4,6,1)$ Explain why each of the two subsets $B_0 = \left\{u_0, u_2,u_3\right\}$ and $B_1 = \left\{u_1, u_2, u_3\right\}$ forms a basis of $\mathbb{R}^3$. If we write $[\mathbf{x}]_0$...

Homework Statement Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question. Consider the...

Homework Statement Homework Equations[/B] The Attempt at a Solution From that point, I don't know what to do. How do I prove linear independence if I have no numerical values? Thank you.

I want to find basis for a+ax^2+bx^4 belong to p4. I am getting the following result is it right? =>a(1+x^2) + b(x^4) => basis ={1+x^2, x^4} Is that right ? Please help me any help is appreciated.

Homework Statement Let U is the set of all commuting matrices with matrix A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...

I want ask another basic question related to this paper - http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922 If I have basis vectors for a curvilinear coordinate system(Euclidean space) that are completely orthogonal to each other(basis vectors will change from point to point)...

Homework Statement What is the area of the primitive cell for the lattice shown below? The nearest neighbor separation is "a." Homework Equations Here's the lattice we were given on our handout, and I have added the lines to indicate the square lattice (in red), the basis (in purple), and...

I'm looking for a nice set of basis matrices ##B_{i,j}## that cover the matrices of size ##n \times n## when linear combinations are allowed. The nice property I want them to satisfying is something like ##B_{i,j} \cdot B_{a,b} = B_{i+a, j+b}##, i.e. I want multiplication of two basis matrices...

Homework Statement Let and are two basis of subspaces and http://www.sosmath.com/CBB/latexrender/pictures/69691c7bdcc3ce6d5d8a1361f22d04ac.png. Find one basis of http://www.sosmath.com/CBB/latexrender/pictures/38d4e8e4669e784ae19bf38762e06045.png and...

From what I've seen so far, the basis of the solution space for all the constant coefficient homo linear DE's have been linear combinations of the exponential function e or of some polynomial multiplied by the exponential function. Is this always true that these DE's always result in solutions...

This is Exercise 2.20 in Nielsen and Chuang's Quantum Computation and Quantum Information, on page 71. Suppose $A'$ and $A''$ are matrix representations of an operator $A$ on a vector space $V$ with respect to two different orthonormal bases, $|v_i\rangle$ and $|w_i\rangle$. Then the elements...

Why isn't the second line in (5.185) ##\sum_k\sum_l<\phi_m\,|\,A\,|\,\psi_k><\psi_k\,|\,\psi_l><\psi_l\,|\,\phi_n>##? My steps are as follows: ##<\phi_m\,|\,A\,|\,\phi_n>## ##=\int\phi_m^*(r)\,A\,\phi_n(r)\,dr## ##=\int\phi_m^*(r)\,A\,\int\delta(r-r')\phi_n(r')\,dr'dr## By the closure...

Hi pf, Having some trouble with basis vectors for expanding a given vector in 3-D space. Any given vector in 3-D space can be given by a sum of component vectors in the form: V = e1V1 + e2V2 + e3V3 (where e1, e2 and e3 are the same as i, j and k unit vectors). Equation 1. I am happy with...

I get the notion that an elementary particle derives from a localized perturbation of that particles quantum field. What I don't get is how that perturbation can lead to two alternative quantum states for that particle - for example, an electron with two spin states (spin up and spin down). Are...

Can you describe or how does one visualize a decoherence branch where the position basis is not preferred (copenhangen, bohmian) but momentum basis (as say one of the Everett). Or if a world has position basis disallowed (suppressed for sake of discussion) and it's all momentum basis.. would...

The wiki page https://en.wikipedia.org/wiki/Bell's theorem states the following which I agree with: Suppose the two particles are perfectly anti-correlated—in the sense that whenever both measured in the same direction, one gets identically opposite outcomes, when both measured in opposite...

Homework Statement Find basis and dimension of V,W,V\cap W,V+W where V=\{p\in\mathbb{R_4}(x):p^{'}(0) \wedge p(1)=p(0)=p(-1)\},W=\{p\in\mathbb{R_4}(x):p(1)=0\} Homework Equations -Vector spaces The Attempt at a Solution Could someone give a hint how to get general representation of a vector...

Homework Statement This is a problem from Haim Brezis's functional analysis book. Homework EquationsThe Attempt at a Solution I'm assuming (e)n is the vectors like (e)1 = (1,0,0), (e)2=(0,1,0) and so on. We know every hilbert space has an orthonormal basis. I also need to know the...

Mod note: Moved from Precalc section 1. Homework Statement Given l : IR3 → IR3 , l(x1, x2, x3) = (x1 + 2x2 + 3x3, 4x1 + 5x2 + 6x3, x1 + x2 + x3), find Ker(l), Im(l), their bases and dimensions. My language in explaining my steps is a little sloppy, but I'm trying to understand the process and...

Homework Statement Find a basis for the following vector space: ## V = \{ p \in \mathbb C_{\leq4} ^{[z]} | \ p(1)=p(i) ## and ## p(2)=0 \} ## (Where ## \mathbb C_{\leq4} ^{[z]} ## denotes the polynomials of degree at most 4) Homework Equations N/A The Attempt at a Solution I tried to find...

$\text{Let } L_{n,i}, i = 0,...,n, \text{be the Lagrange nodal basis at} x_0 < x_1<...<x_n$. Show that, for any polynomial $q \in P_n$ $$\sum_{i=0}^nq(x_i)L_{n,i}(x)= q(x)$$ I don't know how to begin this proof. I know what a lagrange polynomial is, but I am not sure how to begin. If someone...

Hello! (Wave) Find the basic feasible solutions of the system of restrictions: $$2x_1+x_2+x_3=10 \\ 3x_1+8x_2+x_4=24 \\ x_2+x_5=2 \\ x_i \geq 0, i=1,2,3,4,5$$ We notice that the rank of the matrix $A=\begin{pmatrix} 2 & 1 & 1 & 0 & 0\\ 3 & 8 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 1 \end{pmatrix}$ is...

So we obtain the perturbation Hamiltonian H as something proportional to S.L/r3 and the first order energy shift is then the expectation value of this perturbation Hamiltonian in the state that is being perturbed. So let a general gross structure state that we are perturbing be |n l ml s ms >...

I knew that a polynomial of degree n has n+1 basis, i.e 1,x,x^2...x^n; But what if a0=0,i.e the constant term is 0, like x^3+x, then what is the dimension and the basis? Is there only x(one dimension) as the basis?

Hello I am trying to learn linear algebra, and I came across this definition of basis minor on this webpage: https://en.wikibooks.org/wiki/Linear_Algebra/Linear_Dependence_of_Columns "The rank of a matrix is the maximum order of a minor that does not equal 0. The minor of a matrix with the...

I'm stuck on a relation issue if there is a direct relation at all. If I were to verify that a subset is a subspace of a vector space V, would it then be correct to check that subset for linear independence to verify that the subset spans the subspace? I'm not sure if I'm following the...

Homework Statement Assume that (|v_1>, |v_2>, |v_3>) is an orthonormal basis for V. Show that any vector in V which is orthogonal to |v_3> can be expressed as a linear combination of |v_1> and |v_2>.Homework Equations Orthonormality conditions: |v_i>*|v_j> = 0 if i≠j OR 1 if i=j. The Attempt...

I'm trying to understand the maths of QM from Shankar's book - Principles of Quantum Mechanics: On page 21 of that book, there is a general derivation that if we have a relation: |v'> = Ω|v> Where Ω is a operator on |v> transfroming it into |v'>, then the matrix entries of the operator can be...

Hi, I'm doing a first course in GR and have just found out that \eta_{ab} = g(\vec{e}_{a}, \vec{e}_{b}) = \vec{e}_{a} \cdot \vec{e}_{b} where g is a tensor, here taking the basis vectors of the space as arguments. I haven't seen this written explicitly anywhere but does this mean that...

Hi. In GR , covariant differentiation is used because the basis vectors are not constant. But , what about in SR ? If the basis vectors are not Cartesian then they are not constant. Does covariant differentiation exist in SR ? And are for example spherical polar basis vectors which are not...

Homework Statement Determine if the following sets are bases for P_2(R) b) (1+2x+x^2, 3+x^2,x+x^2) d) (-1+2x+4x^2, 3-4x-10x^2,-2-5x-6x^2) Homework Equations Bases IF Linear Independence AND span(Set)=P_2(R) RREF = Reduced Row Echelon Form The Attempt at a Solution My first question here...

What is the dimensionality, N, of the Hilbert space (i.e., how many basis vectors does it need)? To be honest I am entirely lost on this question. I've heard of Hilbert space being both finite and infinite so I'm not sure as to a solid answer for this question. Does the Hilbert space need 4...

Hello everyone, I m currently working on a problem that is freaking me out a bit, suppose I have a second quantized hamiltonian: \begin{eqnarray} H=H_{0}+ \epsilon d^{\dagger}d + V(d{\dagger}c_{0} + h.c) \end{eqnarray} In terms of some new operators, I would like to rotate the hamiltonian, so...

Homework Statement Let ##f : \mathbb{R}^n \rightarrow \mathbb{R}^m## be a linear function. Suppose that with the standard bases for ##\mathbb{R}^n## and ##\mathbb{R}^m## the function ##f## is represented by the matrix ##A##. Let ##b_1, b_2, \ldots, b_n## be a new set of basis vectors for...

< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > Hi! Can help me with this problem with my exercise?I don´t know if i did it okay or i have to do anymore Is there another form to do it? Be the π plane, whose equation with the base B (with...

Hi, sadly my textbook assumes a knowledge I didn't have, of change of basis matrices & coordinate systems for linear transformations; so I have been trolling around the web to fill in the gaps as best I can. I have an open post that has no replies -...

Hi - this follows on from my earlier post - http://mathhelpboards.com/linear-abstract-algebra-14/all-basis-go-standard-basis-16232.html. Just like to confirm my understanding so far ... Theorem Given two vector spaces $ V,W $, a basis $ {α_1 ,…,α_n } $ of V and a set of n vectors $ {β_1 ,…,β_...

I am sure this is very 'basic' :-), I've not read it anywhere - but it's just struck me that every basis in $R^n$ can be row reduced to the std basis for $R^n - (e_1, e_2,...e_n) $ . Row reducing is the way I know to test if a given Matrix of column vectors is a basis. Row reducing also...

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...

Homework Statement From Linear Algebra with applications 7th Edition by Keith Nicholson. Chapter 9.2 Example 2. Let T: R3 → R3 be defined by T(a,b,c) = (2a-b,b+c,c-3a). If B0 denotes the standard basis of R3 and B = {(1,1,0),(1,0,1),(0,1,0)}, find an invertible matrix P such that...

Hello people ! I have been studying Zettili's book of quantum mechanics and found that spherical harmonics are written <θφ|L,M>. Does this mean that |θφ> is a basis? What is more, is it complete and orthonormal basis in Hilbert? More evidence that it is a basis, in the photo i uploaded , in...

I'm having trouble understanding those concepts in the title. Can someone explain those concepts in an easy to understand manner? Please don't refer me to a wikipedia page. I know some linear algebra and multi-variable calculus. Thank you.

I've been reading a book on linear algebra. It talks about finding the the basis of kernel and image of a linear transformation. I understand how to find the basis of the kernel, but I don't understand how to find the basis of the image. Could someone please explain a method of doing it? Thank you!

I'm just learning about the whole Dirac notation stuff and I have come across the fact that bra's and ket's are somehow independent of bases. Or rather that they do not need the specification of a basis. I really don't understand this from a vector point of view. Maybe that is the problem...

Hi, I have a 3d space with metric ds^2= -r^a dt^2 + r^bdr^2 +r^2 dy^2 and I need to construct an orthonormal frame. The first of these three basis vectors is fixed, let's say as e_0=A \partial_t + B \partial_r + C \partial_y To find the other two I set v_1=\partial_t, v_2=\partial_r and...

I think I'm having a really big block over a really small hangup, so forgive me. I'm on an internship and this is my first time working with rotations/change of basis in a real world scenario. I'm creating a calculator that gives torque in a door mechanism, after a given roll and pitch of the...

I Have been reading hartle's book on Gr which states that basis four vectors point in x,y,z,t coordinate axes. So are they similar to the coordinate axis of Cartesian plane.

What is the scalar product of orthonormal basis? is it equal to 1 why is a.b=ηαβaαbβ having dissimilar value

I'm confused by the following passage in our book (translated). An alternative too choosing the normed tangent vectors ##\vec e_i = \frac{1}{h_1}\frac{\partial \vec r}{\partial u_i}## with scale factors ##h_i = \left| \frac{\partial \vec r }{\partial u_i} \right| ## is to choose the normal...