Basis Definition and 1000 Threads

Homework Statement Find bases for the following subspace a of r^3 Y+z=0 The Attempt at a Solution First I found a normal to this plane n=(0,1,1) Then I found two vectors which are orthogonal to the normal u=(0,-1,1), v=(1,0,0) Is this correct the answer in my book has...

Homework Statement Consider the three operators defined by $$\left(S_i\right)_{jk} = -i\epsilon_{ijk}$$ in the x-y-z space and the basis vectors given in x-y-z space as $$e^{\left(1\right)} = -\frac{1}{\sqrt{2}}\left(e_x + ie_y\right), e^{\left(0\right)} = e_z, e^{\left(-1\right)} =...

Homework Statement Note: I am going to use |a> <a| to denote ket and bra vectors The components of the state of a system| ω1> in some basis |δ1>, |δ2>, |δ3> are given by <δ1|ω1> = i/sqrt(3), <δ2|ω1> = sqrt(2/3), <δ3|ω1> = 0 Find the probability of finding the system in the state |ω2>...

Find an orthonormal basis for P2(ℂ) with respect to the inner product: <p(x),q(x)> = p(0)q(0) + p(i)q(i) + p(2i)q(2i) the q(x) functions are suppose to be the conjugates I just don't know how to write it on the computer Attempt: This is where I'm having trouble. So usually I'm given...

Homework Statement write the following in K basis: A=∫|x><x|dx where the integral limits are from -a to a Homework Equations The Attempt at a Solution I tried solving it by inserting the identity I=∫|k><k|dk where the integral limits are from -∞ to +∞ but then I do not...

Homework Statement For my homework assignment, I'm supposed to find a basis for the space of 3x3 matrices that have zero row sums and separately for zero row columns. I am having a hard time with this as it seems to me that there are a lot of combinations I have to consider. For the first...

I have a question regarding an exercise I am doing. It is an electron confined to move on a cylinder and I am asked to: "Find the expectation value of Ly and Lz" in the unperturbed basis. I am just not sure what is meant by the expectation value in a basis? I know what the expectation value is...

Here is the question: Here is a link to the question: Matrix of change of basis? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

In Wald's GR he makes use of a coordinate basis consisting of ∂/∂x^{n} where n runs over the coordinates, and I understand his argument that ∂f/∂x^{n} are tangent vectors, but I can't wrap my head around the operator ∂/x^{n} spanning a tangent space of a manifold. Any clarification on this would...

Hello, are there sets of functions that form an orthonormal basis for the space of square integrable functions over the reals L2(ℝ)? According to Wikipedia the hermite polynomials form an orthogonal basis (w.r.t. to a certain weight function) for L2(ℝ). So I guess it would be a matter of...

I quote un unsolved question posted in MHF on December 8th, 2012 by user bonfire09.

Find a basis for the subspace S = span{(1,2,1,2,1) , (1,1,2,2,1), (0,1,2,0,2)} of Z53 (The set of elements in the field of modulus 3) Attemept: So the issue isn't in finding a basis per say. If this was the field of Real numbers I wouldn't have an issue, I would just row reduce and use the...

[FONT=Lucida Grande]See a picture that represents the relations of the two triangles [FONT=Lucida Grande]what is a "?" [FONT=Lucida Grande]3?3=3 [FONT=Lucida Grande]3?3=4 [FONT=Lucida Grande]3?3=5 [FONT=Lucida Grande]3?3=6 [FONT=Lucida Grande]3?3=7 [FONT=Lucida Grande]3?3=8 [FONT=Lucida...

Hi, I'm learning about vector spaces and I would like to ask some questions about it. Suppose I have a vector space V, and a basis for V \{v_1, ... v_n\}. Then there is a dual space V^* consisting all linear functions whose domain is V and range is ℝ. Then the space V^* has a dual basis \{x_1...

Homework Statement For a lattice with a two atoms basis, the two dispersion relations valid for Ka = ±∏ w2 = 2C/M2 and w2 = 2C/M1 Show that under these conditions the lattice acts as two independent lattices (one lattice per each atom) with one of the lattices moving while the other is...

Throughout mathematics, we see the appearance of pi and e, they are integral to every part of mathematics. could this be because the reason all or at least most mathematical problems arise because of the tension between multiplication and addition (pi and e) itself?. certainly the deepest...

The expansion theorem in quantum mechanics states that a general state of a system can be represented by a unique linear combination of the eigenstates of any Hermitian operator. If that's the case then that would imply we would be able to represent the spin state of a particle in terms of...

Hi. Define a linear mapping F: M2-->M2 by F(X)=AX-XA for a matrix A, and find a basis for the nullspace and the vectorspace(not sure if this is the term in english). Then I want to show that dim N(F)=dim V(F)=2 for all A, A≠λI, for some real λ. F(A)=F(E)=0, so A and E belongs to the nullspace...

My project produces 12 kgm torque so,I need a list of Generator which runs on 12 kgm torque and how much Current it produces... Please advise

Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform. My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where...

Let B={b1,b2} and C={c1,c2} be basis. Then the change of coordinate matrix P(C to B) involves the C-coordinate vectors of b1 and b2. Let [b1]c=[x1] and [b2]c=[y1] ...[x2]...[y2]. Then by definition [c1 c2][x1]=b1 and [c1 c2][y1]=b2. I don't get how you can ....... [x2].....[y2] multiply the...

Homework Statement I have my quantum mechanics final creeping up on me and I just have a question about something that doesn't appear to be covered in the text. Let's say you have a wave function of the following form for a linear harmonic oscillator: \Psi = c_1 | E_1 \rangle + c_2 | E_2...

Homework Statement Problem is assuming the mapping T: P2---->P2 defined by T(a0+a1t+a2t2)=3a0+(5a0-2a1)t+(4a1+a2)t^2 is linear. Find the matrix representation of T relative to Basis B={1,t,t^2}. The part that I am confused on is when I go plug in the basis values T(1),T(t),and T(t^2)? I don't...

Homework Statement Write down a basis for the space of nxn symmetric matrices. The Attempt at a Solution I just need to know what the notation for this sort of thing is. I understand what the basis looks like, and I was even able to calculate that it would have dimension...

Hey! Let M and N be two natural numbers and N>M. I want to build a set A with N vectors of size M such that each subset S of A, where |S| = M, contains linearly independent vectors. Another way to put it is that every S should be a basis for R^M. Any ideas? Thanks!

I have a density matrix in one basis and need to change it to another. I know the eigenvectors and eigenvalues of the basis I want to change to. How do I do this? Any help really appreciated- thanks!

Homework Statement I need to express the rotation operator as follows R(uj) = cos(u/2) + 2i(\hbar) S_y sin(u/2) given the fact that R(uj)= e^(iuS_y/(\hbar)) using |+-z> as a basis, expanding R in a taylor series express S_y^2 as a matrix Homework Equations I know...

Homework Statement (a.) Find an orthonormal basis of R^4 spanned by {1,1,1,1},{1,0,0,1}, and {0,1,0,1}. (b.) Use the inner product to express {2,2,2,2} as a linear combination of the basis vectors. Do not solve the equations. Homework Equations gram schmidt orthogonalization and...

Hello, I am doing calculation on change of basis vector. But I am unable to understand why we do it. I mean to say what is the use of it and where in physics or maths it is used. Can anybody please explain it?

I'm sure you've all heard of Prop 37, but I'll write a short introduction. In the state of California, which is located in the United States, residents can vote on a proposition. That proposition becomes law if they vote in favor of it. Proposition 37 was created in response to the belief...

Why would we want to transform a vector in our normal basis (xyz axes) to another basis? The only situation I can recall is when we are looking at a force applied on an inclined plane. Are there any other real life examples where this may be necessary?

Homework Statement Shanker 1.7.1 3.)Show that the trace of an operator is unaffected by a unitary change of basis (Equivalently, show TrΩ=TrU^{\dagger}ΩUHomework Equations I can show that via Shanker's hint, but I however can't see how a unitary change of basis links to TrΩ=TrU^{\dagger}ΩU...

The problem is attached, I did parts 1-3, but I am having trouble with part 4. This is what i was planning on doing for part 4 (my teacher said this wasn't the correct method): set T(v)=0 and solve the augmented matrix 1 0 -1 1 0 2 1 -2 4 0 3 1 -1 7 0 rref gives 1 0 0 2 0 0 1 0 2 0 0 0 1 1 0...

The problem is attached. I am instructed to find a basis for the nullspace of T.A basis for a 2x2 matrix is 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1Applying the transformation to each of these gives 0 0 0 0 0 2 0 0 0 0 -2 0 0 0 0 0 respectively. Now this is where I get stuck. How do I find a...

The problem is attached. The problem is "find a basis for the range of the linear transformation T." p(x) are polynomials of at most degree 3. R(T)={p''+p'+p(0) of atmost degree 2} This is pretty much as far as I got. I'm not sure how to do the rest. I'm thinking of picking a...

T: P2 → R (the 2 is supposed to be a subscript) The P is supposed to be some weird looking P denoting that it is a polynomial of degree 2. T (p(x)) = p(0) Find a basis for nullspace of linear transformation T.The answer is {x, x^2} I want to make sure I'm interpreting this correctly. It...

This thread is spawned from an earlier one https://www.physicsforums.com/showthread.php?t=647147&page=7 For the stationary ( ie comoving ) frame in the Schwarzschild spacetime the co-basis of the frame field is s_0= \sqrt{\frac{r-2m}{r}}dt,\ \ s_1=\sqrt{\frac{r}{r-2m}}\ dr,\ \ s_2=r\...

Suppose that S = {v1, v2, v3} is a basis for a vector space V. a. Determine whether the set T = {v1, v1 + v2, v1 + v2 + v3} is a basis for V. b. Determine whether the set W = {−v2 + v3, 3v1 + 2v2 + v3, v1 − v2 + 2v3} is a basis for V. I must check if they're linearly independent...

The problem is attached. I'm having problems with parts a and c, well maybe not part a (probably just need to check if I did this part right. I'm just not sure if I'm wording part a right. Anyways for part a I must prove it's a subspace so I must satisfy 3 conditions: 1) 0 is in S 2) if U and...

I know how to do the problem, just put the 4 vectors in matrix form and find for what values of k is the detminant =0. the answer is then that k can't equal the value that was found. Is there a easier way to do this? My method involves finding the determinant using the expansion method...

Show that if S = {v1, v2, . . . , vn} is a basis for R
n and A is an n × n invertible matrix, then S' = {Av1,Av2, . . .,Avn} is also a basis. I need to show that: 1) Av1, Av2,...Avn are linearly independent 2) span(S)=Rn I'm having some problems with this. I see that S'=AS (duh)...

I'm not sure how to start this problem. All i know is a diagonal matrix consists of all 0 elements except along the main diagonal. But how do I even find a basis for this?

Hello I have this problem, I find it difficult, any hints will be appreciated... Two subspaces are given (W1 and W2) from the vector space of matrices from order 2x2. W1 is the subspace of upper triangular matrices W2 is the subspace spanned by...

According to my professor, there exist infinite dimensional vector spaces without a basis, and he asked us to find one. But isn't this impossible? The definition of a dimension is the number of elements in the basis of the vector space. So if the space is infinite-dimensional, then the basis...

Homework Statement The problem is Exercise 2 in the picture http://postimage.org/image/3ou3x1sh7/ Homework Equations The hint says: can you express and three-dimensional vector in terms of just two linearly independent vectors? The Attempt at a Solution I have no idea where...

Homework Statement Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B. Homework Equations The Attempt at a...

Hi! There is a concept I don't understand and would love to have is cleared... What is the meaning of a lattice with a basis? What do I need it for? Say I have a honeycomb structure. (fig 1) and a basis as mentioned there (did I understand it right? is it the basis? ) why does it become...

When you have a matrix like: 3 1 0 1 The RREF is 1 0 0 1 the identity matrix. Is the basis always [0 0]^t?

Suppose that T1: V → V and T2: V → V are linear operators and {v1, . . . , vn} is a basis for V . If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n, show that T1(v) = T2(v) for all v in V . I don't understand this question. They said If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n...

Homework Statement The set K of 2 × 2 real matrices of the form [a b, -b a] form a field with the usual operations. It should be clear to you that M22(R) is a vector space over K. What is the dimension of M22(R) over K? Justify your answer by displaying a basis and proving that the set...