Basis Definition and 1000 Threads

Hi everyone, :) I have a little trouble understanding what Canonical basis means in the following question. I thought that Canonical basis is just another word for the Standard basis. Hope you people could clarify the difference between these two in the given context. :) Question: Find the...

Hi everyone, :) Here's a question with my answer. It's pretty simple but I just want to check whether everything is perfect. Thanks in advance. :) Question: Let \(f:\,\mathbb{C}^2\rightarrow\mathbb{C}^2\) be a linear transformation, \(B=\{(1,0),\, (0,1)\}\) the standard basis of...

Homework Statement There is a standard basis, B = (1; z; z^2; z^3; z^4) where B is the basis of a R4[z] of real polynomials of at most degree 4. I need to find another basis B' for R4[z] such that no scalar multiple of an element in B appears as a basis vector in B' and also prove that...

Edit complete, but it doesn't seem as though I can change the title. The latex arrows next to the 'P' aren't showing up for me but they're supposed to be left arrows Homework Statement Let B and C be bases of R^2. Find the change of basis matrices P_{B \leftarrow C} and P_{C\leftarrow B}...

The following is a question regarding the derivation of Einstein's field equations. Background In deriving his equations, it is my understanding that Einstein equated the Einstein Tensor Gμv and the Cosmological Constant*Metric Tensor with the Stress Energy Momentum Tensor Tμv term simply...

My book says that "the countability of the ONS in a hilbert space H entails that H can be represented as closure of the span of countably many elements". I must admit my english is probably not that good. At least the above quote does not make sense to me. What is it trying to say? Previously...

I have been reading an introductory book to General Relativity by H Hobson. I have been following it step by step and now I am stuck. It is stated in the book that: "It is straightforward to show that the coordinate and dual basis vectors themselves are related... "ea = gabeb ..." I have...

Homework Statement A function F(x) = x(L-x) between zero and L. Use the basis of the preceding problem to write this vector in terms of its components: F(x)= \sum_{n=1}^{\infty}\alpha _{n}\vec{e_{n}} If you take the result of using this basis and write the resulting function outside the...

So the other day in class my teacher gave a proof for the completeness of \phi_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx} in L^2([-\pi,\pi]) . And I'm trying to convince my self I understand it at least a little. He defined Frejer's Kernel K_n(x) = \frac{1}{2\pi(n+1)}...

Hi everyone, :) We are given the following question. I don't expect a full answer to this question, but I don't have any clue as to what is a Matrix Unit. Do any of you people know what is a matrix unit?

[b]1. The problem statement find the β coordinates ([x]β) and γ coordinates ([x]γ) of the vector x = \begin{pmatrix}-1\\-13\\ 9\\ \end{pmatrix} \in\mathbb R if {β= \begin{pmatrix}-1\\4\\ -2\\ \end{pmatrix},\begin{pmatrix}3\\-1\\ -2\\ \end{pmatrix},\begin{pmatrix}2\\-5\\ 1\\ \end{pmatrix}}...

Homework Statement Can you find a basis {p1, p2, p3, p4} for the vector space ℝ[x]<4 s.t. there does NOT exist any polynomials pi of degree 2? Justify fully.Homework Equations The Attempt at a Solution We know a basis must be linearly independant and must span ℝ[x]<4. So intuitively if there...

I'm having some set theoretic qualms about the following argument for the following statement: Let V be a vector space of dimension n and let S be a generating set for V. Prove that some subset of S is a basis for V. The argument is as follows: If ##V = \{ 0 \} ## then it is trivial...

Let c_00 be the subspace of all sequences of complex numbers that are "eventually zero". i.e. for an element x∈c_00, ∃N∈N such that xn=0,∀n≥n. Let {e_i}, i∈N be the set where e_i is the sequence in c_00 given by (e_i)_n =1 if n=i and (e_i)_n=0 if n≠i. Show that (e_i), i∈N is a basis for...

Hello, Find a basis for subspace in $$P_3(\mathbb{R})$$ that containrar polynomial $$1+x, -1+x, 2x$$ Also the hole ker T there $$T: P_3(\mathbb{R})-> P_3(\mathbb{R})$$ defines of $$T(a+bx+cx^2+dx^3)=(a+b)x+(c+d)x^2$$ I am unsure how to handle with that ker.. I am aware that My bas determinant...

Homework Statement Letting u=[3, 0, -5], v=[2, 1, 5] and w=[-1, 3, 4], how would I show that a general vector can be written as a linear combination of this 'basis?' Without using an augmented matrix and getting a really messy result by using arbitrary a, b, and c values as the solutions...

Hello, I was wondering if it is allowed when doing problems on multiple unit mass balances to switch the basis made at the beginning of the problem and apply it to a new control volume, while keeping the old basis for previous control volumes? Thank you

Homework Statement Let L: R2 → Rn be a linear mapping. We call L a similarity if L stretches all vectors by the same factor. That is, for some δL, independent of v, |L(v)| = δL * |v| To check that |L(v)| = δL * |v| for all vectors v in principle involves an infinite number of...

Homework Statement ##S## is a linear transformation and ##\{u_{1},u_{2}\}## is a basis for the vector space. $$ S(u_{1})=u_{1}+u_{2}\\ S(u_{2})=-u_{1}-u_{2} $$ I would like to find a basis of the null space and range of ##S##.Homework Equations In my text, it says that the proper matrix...

Let ##T: V → V ## be a linear map on a finite-dimensional vector space ##V##. Let ##W## be a T-invariant subspace of ##V##. Let ##γ## be a basis for ##W##. Then we can extend ##γ## to ##γ \cup S##, a basis for ##V##, where ##γ \cap S = ∅ ##, so that ## W \bigoplus span(S) = V ##. My question...

Hello! I am stuck at the following exercise: "Construct an orthogonal basis of R^{3} (in terms of Euclidean inner product) that contains the vector \begin{pmatrix}2\\1 \\-1 \end{pmatrix} " What I've done so far is: Let {(a,b,c), (k,l,m), (2,1,-1)} be the basis. Then since the basis has to...

Homework Statement Hi guys, actually this isn't a homework question, but rather part of the working in a textbook on Linear Algebra. Homework Equations The Attempt at a Solution I'm not sure why it's U*li instead of U*il. Shouldn't you flip the order when you do a matrix...

For example, if were given only a vector <5, 3, 1>, is this assumed to be respect with the standard basis of R^3? And would this mean that any nonstandard basis is with respect to a standard basis?

Homework Statement Find a basis for the following vector space: The set of 2x2 matrices A such that CA=0 where C is the matrix : 1 2 3 6The Attempt at a Solution I multiplied C by a general 2x2 matrix ...

I keep reading about polar unit vectors, and I am a bit confused by what they mean. In the way I like to think about it, the n-tuple representation of a vector space is just a "list" of elements from the field that I have to combine (a.k.a. perform multiplication) with the n vectors in some...

Is there a hole in knowledge as to the origins of PDEs? If there is a void, is Schwartz space a suitable basis? Schwartz spaces are intermediate between general spaces and nuclear spaces. Infra-Schwartz spaces are intermediate between Schwartz spaces and reflexive spaces.

In my linear algebra class we previously studied how to find a basis and I had no issues with that. Now we are studying the basis of a row space and basis of a column space and I'm struggling to understand the methods being used in the textbook. The textbook uses different methods to find these...

In RQM all systems are observers. Select the viewpoint with a system S and an observer O. The systema has 2 eigenfunctions |0> and |1> in a basis. Then the evolution from |init>_{O}(|0>+|1>)_{S}. Then the system evolutions to |O0>|0>_{S} or to |O1>|1>_{S} . But how does the measurement select...

If I have a 45 degree linear polarized light which I then circularly polarize using a 1/4 wave plate and put this through an optical rotary crystal and then using the equivalent 1/4 wave plate but in the reverse oriention, will I get back a 45 degree linear polarized light? Put another way...

Essentially, I have to show that where [SIZE="3"]{e_1,...,e_n} forms the basis of [SIZE="3"]L, no family of vectors [SIZE="3"]{e'_1 ,..., e'_m} with [SIZE="3"]m>[SIZE="3"]n can serve as the basis of [SIZE="3"]L. The book shows this by saying there exists a [SIZE="3"]0 vector such that...

Let S be a subspace of $\mathbb{R^2}$, such that $S=\{(x,y):2x+3y=0 \}$. Find a basis,$B$, for $S$ and write $u=(-9,6)$ in the $B$ basis. So, I started to solve $2x+3y=0$ for $x$ and I got $x=-\frac{3}{2}y$. Then I could write, $\left[ \begin{matrix} x \\ y \end{matrix}\right] = \left[...

Homework Statement Let H be the set of all vectors of the form (a-3b, b-a, a, b) where a and b are arbitrary real scalars. Show that H is a subspace of ℝ^4 and find a basis for it. Right, I've shown it's a proper subspace, just need help with finding a basis. Is {a-3b, b} a suitable...

I am reading Dummit and Foote Section 9.6 Polynomials In Several Variables Over a Field and Grobner Bases I have a very basic question regarding the beginning of the proof of Hilbert's Basis Theorem (see attachment for a statement of the Theorem and details of the proof) Theorem 21...

Hi, Can someone double check I understand this correctly? The turbofan has lower specific fuel consumption because a gas's momentum is proportional to its velocity, whereas a gas's kinetic energy is proportional to its squared velocity. Therefore a turbojet can be made more efficient by adding...

Homework Statement In ##E^3##, given the orthonormal basis B, made of the following vectors ## v_1=\frac{1}{\sqrt{2}}(1,1,0); v_2=\frac{1}{\sqrt{2}}(1,-1,0); v_3=(0,0,1)## and the endomorphism ##\phi : E^3 \to E^3## such that ##M^{B,B}_{\phi}##=A where (1 0 0) (0 2 0) = A (0 0 0)...

I'm having trouble finding the spanning set and basis for the matrix; | a b | | c d | with condition that b=d I'm thinking thinking the spanning set would be A= x B = y C = z Such that x,y,z are all reals, but I can't think of how to find a basis for this, I'm thinking of doing...

Here's the introduction of the paper by Freidel and Hnybida. Quantum geometry is built up of chunks of geometry that contain information relating to volume, areas, angles made with neighbor chunks, etc. The Hilbert space that these chunks (called intertwiners) live in needs a set of basis...

I have been thinking about it for sometime but couldn't really get the answer. This is the progress I have made till now. E |ψ> = H |ψ> E <p|ψ> = <p|H|ψ> Now how to evaluate the number <p|H|ψ>? Although I can evaluate this number by introducng identity operator 1 = \int|x><x| dx. But for...

So, I know that for a set of vectors to be a basis the set of vectors must be linearly independent and also must be a spanning set of vectors. So, they can't be parallel. I still feel that I'm not fully understanding what a basis is. Could someone explain to me, maybe with an example, what is a...

Homework Statement Given matrix A= {[39/25,48/25],[48/25,11/25]} find the basis for both eigenvalues. Homework Equations The Attempt at a Solution I row reduced the matrix and found both eigenvalues. I found λ = -1, and λ = 3. Then, I used diagonalization method [-1I2 - A 0]...

Hi, I'm having trouble understanding the purpose of using two basis in a linear transformation. My lecturer explained that it was a way to find a linear transformation that satisfied either dimension, but I'm having trouble understanding how that relates to the method in finding this...

Google has not been very useful, and Kittel has too little on crystallography. Actually, what's a good source on crystallography?

Homework Statement ##\phi## is an endomorphism in ##\mathbb{E}^3## associated to the matrix (1 0 0) (0 2 0) =##M_{\phi}^{B,B}##= (0 0 3) where B is the basis: B=((1,1,0),(1,-1,0),(0,0,-1)) Find an orthonormal basis "C" in ##\mathbb{E}^3## formed by eigenvectors of ##\phi## The...

I'm working on a problem where I want to write the operator S_z down in terms of some operator(s) in the \vec{J} = \vec{L} + \vec{S} basis so that I can operate S_z on the states \mid \ell, s=1/2, j= \ell\pm1/2, m\rangle but I'm having trouble finding the correct combination of operators...

Hi guys, I'm back and have another Linear Albgera question! Thanks in advance. No idea how to start

Quantum mechanics says that physical observables are self-adjoint operators. Is this correspondence a bijection, ie can we realize any such operator as a physical observable? There are obvious practical concerns with physically realizing certain contrived operators. But are there any...

Does anybody know how to create a orthonormal basis, i.e. a matrix containing orthogonal vectors of norm 1, out of a given direction (normalised vector or versor) in a space with dimension N>3? With "out of a given direction", I mean that the resulting basis would have the first vector equal...

I've done most of this question apart from the very last bit. I have an answer to the very last bit, but it doesn't use any of my previously proved statements and I think they probably mean me to deduce from what I already have. Homework Statement Let V be the finite-dimensional vector...

Firstly; is there a difference between the "regular" polar coordinates that use \theta and r to describe a point (the one where the point (\sqrt{2}, \frac{\pi}{4}) equals (1, 1) in rectangular coordinates) and the ones that use the orthonormal basis vectors \hat{e}_r and...

Homework Statement Suppose we have a system and that {|a>, |b>, ...} is a complete and orthonormal basis for the system Am i right in thinking Ʃ(j) <k|j><j|i> = <k|i> = 0 unless k=i? In other words, does the LHS expression equal the middle one because Ʃ(j) |j><j| is just the insertion...