Basis Definition and 1000 Threads

Homework Statement For the orthonormal coordinate system (X,Y) the metric is \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} Calculate G' in 2 ways. 1) G'= M^{T}*G*M 2) g\acute{}_{ij} = \overline{a}\acute{}_{i} . \overline{a}\acute{}_{j} Homework Equations \begin{pmatrix}...

Hey guys so we need to find the basis for 0 1 \sqrt{2} 0 0 0 0 0 0 I know how you do it. But my prof says that one of the basis vectors is (1 0 0) but I don't know how he arrives at this?

Hello all, I don't have a question on homework specifically, but I need clarification on something I'm reading in the textbook. I will be starting an abstract algebra class in the spring and it's been quite a few years since I've had linear algebra, so I'll be reviewing that material before the...

Homework Statement Given ##S = \{1, x, x^2\}##, find the coordinates of ##x^2 + x + 1## with respect to the orthogonal set of S.Homework Equations Inner product on polynomial space: ##<f,g> = \int_{0}^{1} fg \textrm{ } dx## The Attempt at a Solution I used Gram-Schmidt to make ##S## orthogonal...

Hi everyone, :) This seems like a pretty simple question, but up to now I haven't found a method to solve it. Hope you can provide me a hint. :) Problem: Let \(V\) be a space with basis \(B=\{b_1,\,b_2,\,b_3,\,b_4,\,b_5\},\,U\) the subspace spanned by \(u_1=b_1+b_2+b_3+b_4+b_5\)...

Let n be a positive integer, and for each $j = 1,..., n$ define the polynomial $f_j(x)$ by f_j(x) = $\prod_{i=1,i \ne j}^n(x-a_i)$ The factor $x−a_j$ is omitted, so $f_j$ has degree n-1 a) Prove that the set $f_1(x),...,f_n(x)$ is a basis of the vector space of all polynomials of degree ≤ n -...

Homework Statement Find a basis for the subset S = {(1, 2, 1), (2, 1, 3), (1, -4, 3)} Homework Equations The Attempt at a Solution I'm not absolutely sure I'm doing this correctly but here is my attempt: First, I put the vectors in S in the rows of a matrix (using multiple...

Hi everyone, :) I am trying to find an approach to solve this but yet could not find a meaningful one. Hope you can give me a hint to solve this problem. Problem: Prove that for any bivector \(\epsilon\in\wedge^2(V)\) there is a basis \(\{e_1,\,\cdots,\,e_n\}\) of \( V \) such that...

Hello everyone Can anyone help me to solve the following problem Prove that for any bivector $v\in \Lambda^2 (V)$ there is a basis $\{e_1,e_2,...,e_n\}$ of $V$ such that $v=e_1\Lambda e_2 +e_3\Lambda e_4 +...+e_{k-1}\Lambda e_k$ I did it in this way, since the out product $e_i\Lambda e_j$...

The topology ## T ## on a set ## X ## generated by a basis ## B ## is defined as: T=\{U\subset X:\forall\ x\in U\ there\ is\ a\ \beta\in B:x\in \beta \ and\ \beta\subset U \}. But if ##U## is the empty set, and there has to be a ## \beta ## in ##B## that is contained in ##U##, the empty set...

Let β be a basis for ℝ over Q (the set of all rational numbers) and let a\inℝ, a≠1. Show that aβ={ay|y\inβ} is a basis for ℝ over Q for all a≠0. So I need to show (1) Linear independence, and (2) spanning. I am a little confused, especially because the dimension for the vector space is...

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 \neq0 well I did...

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

Hi! I'm having problems understanding the last step of a derivation for a version of a theorem of Gromov's we had in class: In short, the proof takes the Riemannian universal covering (\tilde M, \tilde g, \tilde p) of (M, p) and uses a short basis (\gamma_1, \gamma_2, ...) of the deck...

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...

Given a basis for spacetime ##\{e_0, \vec{e}_i\}## for which ##\vec{e}_0## is a timelike vector. Of these vectors one can make a new basis for which all vectors are orthogonal to ##\vec{e}_0##. I.e. the vectors $$\hat{\vec{e}}_i = \vec{e}_i - \frac{\vec{e}_i \cdot \vec{e}_0}{\vec{e}_0 \cdot...

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 A vector is a geometrical object which doesn't depend on the basis we use to represent it, only its components will change. We can express this by \vec{A}=ƩA_i \hat{ε_i} = Ʃ\tilde{A_i} \vec{ε_i}, where it has been emphasized that the basis ε is not necessarily orthonormal...

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 ...

According to Wikipedia bornology is the minimum amount of structure needed to address boundedness. Topology is the minimum amount of structure needed to address continuity. Topology-bornology (cf. Bornologies and Fuctional Analysis, by Hogbe-Nlend) can be applied to Distributions as bounded...

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 {e_1,...,e_n} forms the basis of L, no family of vectors {e'_1 ,..., e'_m} with m>n can serve as the basis of L. The book shows this by saying there exists a 0 vector such that 0 = \sum_{i=1}^{m}x_ie'_i, where not all x_i vanish. I wanted to show it by...

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 the Proof of Hilbert's Basis Theorem in Rotman's Advanced Modern Algebra ( See attachment for details of the proof in Rotman). Hilbert's Basis Theorem is stated as follows: (see attachment) Theorem 6.42 (Hilbert's Basis Theorem) If R is a commutative noetherian ring, the R[x] is...

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...