Basis Definition and 1000 Threads

$\tiny{370.14.2.}$ For the matrix $A=\left[ \begin{array}{rrrr} 1&0&1\\0&1&3 \end{array}\right]$ find a basis for NS(A) and $\dim{NS(A)}$ ----------------------------------------------------------- altho it didn't say I assume the notation means Null Space of A Reducing the augmented matrix...

Verify that $\beta=\left\{\begin{bmatrix} 0\\2 \end{bmatrix} ,\begin{bmatrix} 3\\1 \end{bmatrix}\right\}$ is a basis for $\Bbb{R}^2$ Then for $v=\left[ \begin{array}{c}6\\8\end{array} \right]$, find $[v]_\beta$ ok, I presume next is...

The first part I'm fairly sure is just the regular gradient in polar coordinates typically encountered: $$\nabla u= \hat {\mathbf e_r} \frac {\partial u} {\partial r} + \hat {\mathbf e_\theta} \frac 1 r \frac {\partial u} {\partial \theta}$$ or in terms of scale factors: $$=\sum \hat...

Is this correct? If not any hints on how to find Many thanks

Wet basis 0.75mol C4H10 Requires 4.875 mols O2 Produces 3 mols of CO2 and 3.75 mols of H2O 0.1mol C3H8 Requires 0.5 mols O2 Produces 0.3 mols of CO2 and 0.4 mols of H2O 0.15mol C4H8 Requires 0.9 mols O2 Produces 0.6 mols of CO2 and 0.6 mols of H2O Theoretical oxygen= 6.3mol +10% excess...

If I have a composite system, like a two particle system, for exemple, I can construct my Hilbert space as the tensor product of the hilbert spaces of these particles, and, if ##\{|A;m \rangle \}## and ##\{|B;n \rangle \}## are basis in these hilbert spaces, a basis in the total hilbert space is...

I had assumed that we had to put our values into a matrix so I did [1 2 -1 0; 1 -5 0 -1] and then I would do a=[1; 1] and repeat for b, c, and d. This is incorrect however. I also thought that it could be {(1, 2, -1, 0),(1, -5, 0, -1)} however this was not the answer, and I am unsure of what do...

ok I am new to this basis of kernel and tried to understand some other posts on this but they were not 101 enough Find the basis for kernel of the differential operator $D:C^\infty\rightarrow C^\infty$, $D^4-2D^3-3D^2$ this can be factored into $D^2(D-3)(D+1)$

Hi PF Given some linear differential operator ##L##, I'm trying to solve the eigenvalue problem ##L(u) = \lambda u##. Given basis functions, call them ##\phi_i##, I use a variational procedure and the Ritz method to approximate ##\lambda## via the associated weak formulation $$\langle...

We define the application $T:P_2\rightarrow P_2$ by $$T(p)=(x^2+1)p''(x)-xp'(x)+2p'(x)$$ 1. Give the matrix $\displaystyle\left[T\right]_\infty^\infty$ in the standard basis $\alpha=(x^2,x,1)$ 2 Give the matrix $\displaystyle\left[T\right]_\infty^\infty$ where...

I am working through a book with my professor and we read a section on the dual space, $V^*$. It gives the basis dual to the basis of $V$ and proves that this is in fact a basis for $V^*$. Characterized by $\alpha^i(e_j)=\delta_j^i$ I understand the proof given. But he said a different...

Let $$\left[\begin{array}{rrrrrrr} 1 & 0 & -1 & 0 & 1 & 0 & 3\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 4 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \end{array}\right]$$ Find a basis for the null space of A, the dimension of the null space of A, and...

verify that $\beta =\left\{ \left[\begin{array}{c}0 \\ 2 \end{array}\right], \left[\begin{array}{c}3 \\ 1\end{array}\right]\right\}$ is a basis for $\Bbb{R}^2$ for $v=\left[\begin{array}{c}6\\ 8 \end{array}\right]$ find $|v|_{\beta-}$ ok $x_2=2$ and $x_1=3$ not sure how to answer the rest

Hello all. I'm using Griffiths' Introduction to Quantum Mechanics (3rd ed., 2018), and have come across what, on the face of it, seems a fairly straightforward principle, but which I cannot justify to myself. It is used, tacitly, in the first equation in the following worked example: The...

Hi, I am really new in understanding second quantization formalism. Recently I am reading this journal: https://dash.harvard.edu/bitstream/handle/1/8403540/Simulation_of_Electronic_Structure.pdf?sequence=1&isAllowed=y In brief, the molecular Hamiltonian is written as $$\mathcal{H}=\sum_{ij}...

I am learning the basics of differential geometry and I came across tangent vectors. Let's say we have a manifold M and we consider a point p in M. A tangent vector ##X## at p is an element of ##T_pM## and if ##\frac{\partial}{\partial x^ \mu}## is a basis of ##T_pM##, then we can write $$X =...

Given a basis of a vector space $(V,O_1,O_2)$ can it represent two different non-isomorphic graphs.Any other inputs kind help. It will improve my knowledge way of my thinking. Another kind help with this question is suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the...

Hi, In this presentation about quantum optics it is mentioned that the same quantum state |Ψ> has different expressions in different mode bases : factorized state or entangled state. This presentation is related to this video : In some way entanglement isn't intrinsic. It depend on the...

My question is given an orthonormal basis having the basis elements Ψ's ,matrix representation of an operator A will be [ΨiIAIΨj] where i denotes the corresponding row and j the corresponding coloumn. Similarly if given two dimensional harmonic oscillator potential operator .5kx2+.5ky2 where x...

Homework Statement [/B] A=##\begin{bmatrix} 1 & 2 & 3 \\ 4 & 8 & 12 \end{bmatrix}## Question: Determine a basis for ker(A).Homework Equations Ker(A)=##\begin{bmatrix} 1 & 2 & 3& |&0 \\ 4 & 8 & 12& |&0 \end{bmatrix}## (And reduce with gaussian elimination) The Attempt at a Solution (Sorry...

Hi, I’m just wondering about this: Are there any theoretical reasons why physical laws take the form of 2nd order (in time) differential equations? Or is it just observed to be that way? Are there ANY laws (even in a limited context) which are 3rd (or higher) order in time?

Hey! :o Let $V$ be a vector space with with a 5-element basis $B=\{b_1, \ldots , b_5\}$ and let $v_1:=b_1+b_2$, $v_2:=b_2+b_4$ and $\displaystyle{v_3:=\sum_{i=1}^5(-1)^ib_i}$. I want to determine all subsets of $B\cup \{v_1, v_2, v_3\}$ that form a basis of $V$. Are the desired subsets the...

Hey! :o Let $V$ be a vector space. Let $b_1, \ldots , b_n\in V$ and let $\displaystyle{b_k':=\sum_{i=1}^kb_i}$ for $k=1, \ldots , n$. I want to show that $\{b_1, \ldots , b_n\}$ is a basis of $V$ iff $\{b_1', \ldots , b_n'\}$ is a basis of $V$. I have done the following: Let $B:=\{b_1...

Anyone know how to change a basis of a qubit state of bloch sphere given a general qubit state? There are 3 different basis corresponding to each direction x,y,z where |1> ,|0> is the z basis, |+>, |-> is the x basis and another 2 ket notation for y basis. Given a single state in the x basis...

Hi PF! I'm somewhat new to the concept of completeness, but from what I understand, a particular basis function is complete in a given space if it can create any other function in that space. Is this correct? I read that the set of polynomials is not complete (unsure of the space, since Taylor...

Homework Statement Consider the real-vector space of polynomials (i.e. real coefficients) ##f(x)## of at most degree ##3##, let's call that space ##X##. And consider the real-vector space of polynomials (i.e. real coefficients) of at most degree ##2##, call that ##Y##. And consider the linear...

Is it possible to expand a state vector in a basis where the basis vectors are not eigenvectors for some observable A? Or must it always be the case that when we expand our state vector in some basis, it will always be with respect to some observable A?

Homework Statement I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question? Standard basis vectors: e_1, e_2 or i,j

Why are the limits as so for the integral?

hi. if I know how to convert coordinates from a system to cartesian system, then how can I find basevectors of that coordinatesystem? Is it possible that basevectors are different in different points(with different coordinates)? What is most general definition of basevectors? I tought it would...

Hello, I've a fundamental question that seems to keep myself confused about the mathematics of quantum mechanics. For simplicity sake I'll approach this in the discrete fashion. Consider the countable set of functions of Hilbert space, labeled by i\in \mathbb{N} . This set \left...

What is the history of the concept that a measurement process is associated with a linear opeartor? Did it come from something in classical physics? Taking the expected value of a random variable is a linear operator - is that part of the story?

nmh{796} $\textsf{Suppose $Y_1$ and $Y_2$ form a basis for a 2-dimensional vector space $V$ .}\\$ $\textsf{Show that the vectors $Y_1+Y_2$ and $Y_1−Y_2$ are also a basis for $V$.}$ $$Y_1=\begin{bmatrix}a\\b\end{bmatrix} \textit{ and }Y_2=\begin{bmatrix}c\\d\end{bmatrix}$$ $\textit{ then }$...

Hey! :o Let $K$ be a field and $V$ a $n$-dimensional $K$-vector space with basis $B=\{b_1, \ldots , b_n\}$. $V^{\star}$ is the dual space of $V$. $B^{\star}$ is the dual basis corresponding to $B$ of $V^{\star}$. Let $C=\{c_1, \ldots , c_n\}$ be an other basis of $V$ and $C^{\star}$ its...

Hello! (Wave) Let linear map $f: \mathbb{R}^3 \to \mathbb{R}^2$, $B$ basis (unknown) of $\mathbb{R}^3$ and $c=[(1,2),(3,4)]$ basis of $\mathbb{R}^2$. We are given the information that $cf_s=\begin{pmatrix} 1 & 0 & 1\\ 2 & 1 & 0 \end{pmatrix}$. Let $v \in \mathbb{R}^3$, of which the coordinates...

Hey! :o Let $A\in \mathbb{C}^{2\times 2}$ and $L_A:\mathbb{C}^{2\times 2}\rightarrow \mathbb{C}^{2\times 2}, \ X\mapsto A\cdot X$. We consider the matrix \begin{equation*}A=\begin{pmatrix}-1 & 2 \\ 2 & -4\end{pmatrix}\end{equation*} and the basis \begin{equation*}B=\left \{\begin{pmatrix}1...

[solved] Basis for set of solutions for linear equation Hi, I have this problem I was working through, but I'm not sure that I've approached it from the right way. The problem consists of 3 parts, which build off of each other. I'm pretty confident about the first two parts, but no so much...

Homework Statement Find orthonormal basis for 1, x, x^2 from -1 to 1. Homework Equations Gram-Schmidt equations The Attempt at a Solution I did the problem. My attempt is attached. Can someone review and explain where I went wrong? It would be much appreciated.

Equation 9.2.25 defines the inner product of two vectors in terms of their components in the same basis. In equation 9.2.32, the basis of ## |V \rangle## is not given. ## |1 \rangle ## and ## |2 \rangle ## themselves form basis vectors. Then how can one calculate ## \langle 1| V \rangle ## ? Do...

Homework Statement I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space. Homework EquationsThe Attempt at a Solution The dot product of a vector a with itself can be given by I a I2. Does that expression only apply...

Hi PF! I'm working with some basis functions ##\phi_i(x)##, and they get out of control big, approximately ##O(\sinh(12 j))## for the ##jth## function. What I am doing is forcing the functions to zero at approximately 3 and 3.27. I've attached a graph so you can see. Looks good, but in fact...

Hello! I read about this in several place, but I haven't found a really satisfying answer, so here I am. As far as I understand, non-coordinate basis are mainly obtained from coordinate basis, by making the system orthonormal. For example the unit vector in polar coordinates in the direction of...

While writing down the basis for SU(2), physicists often choose traceless hermitian matrices as such, often the Pauli matrices. Why is this? In particular why traceless, and why hermitian?

Hi all, I'm trying to find a mathematical way of showing that given a complete set $$\left |a_i\right \rangle_{i=1}^{i=dim(H)}∈H$$ together with the usual property of $$\left |\psi\right \rangle = ∑_i \left \langle a_i\right|\left |\psi\right \rangle\left |a_i\right \rangle ∀ \left...

I have a quick, general, query. When conducting an experiment with entangled photons, measuring in the +/- basis would mean in the 45/135 basis?

Homework Statement I am unsure as to how the partial derivative of the basis vector e_r with respect to theta is (1/r)e_theta in polar coordinates Homework EquationsThe Attempt at a Solution differentiating gives me -sin(theta)e_x+cos(theta)e_y however I'm not sure how to get 1/r.

ei=i+j+2vk , how to find the dual basis vector if the above is a natural base?

This came up in another thread. GR more or less follows directly from Lovelock's Theorem. You simply assume the metric has a Lagrangian. Where does that leave other things like the Equivalence principle? Thanks Bill

I read from this page https://properphysics.wordpress.com/2014/06/09/a-no-nonsense-introduction-to-special-relativity-part-6/ that the basis vectors are the canonical basis vectors in any coordinate system. This seems to be wrong, because if that was the case the metric would be the identity...

Homework Statement Express a polynomial in terms of the basis vectors. {x2 + x, x + 1, 2} Homework Equations 3. The Attempt at a Solution [/B] I think the answer is: (x2+x)^2 + (x + 1) + 2 = 0 simplified to become: x4 + 2x3 + x2 + x + 3 = 0