Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Problem 2

Are the linear combinations \(2 x-y-z, 2 y-x-z\) \(2 z-x-y\) linearly independent?

Short Answer

Expert verified
The vectors \(2 x-y-z, 2 y-x-z, 2 z-x-y\) are linearly dependent.

Step by step solution

01

Set Up a Matrix with the Given Vectors

The first step is to set up a matrix using the coefficients of the vectors. This will result in the following matrix A:\n\[ A = \begin{bmatrix} 2 & -1 & -1 \ -1 & 2 & -1 \ -1 & -1 & 2 \end{bmatrix} \]
02

Calculating the determinant

To check for linear independence, compute the determinant of the matrix. If the determinant is nonzero, the vectors are linearly independent. Calculate the determinant (det(A)), using the co-factor method or Sarrus' rule. \n\[ det(A) = 2((2*2 - (-1*-1)) - (-1)(-1*2 - (-1*-1)) + (-1)(-1*-1 - 2*-1))\] \[ = 2(4 -1) - (-1)(-2-1) + (-1)(-1 -2) \]\[ = 2(3) - (-1*-3) - 3\] \[ = 6 -3 -3 = 0 \]
03

Check the determinant

Check the computed determinant of A. If it equals to zero, this means the vectors are linearly dependent. The determinant we found is 0.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate the expectation values of the operators \(p_{x}\) and \(p_{x}^{2}\) for a particle with wavefunction \((2 / L)^{1 / 2} \sin (\pi x / L)\) in the range 0 to \(L\)

Evaluate the commutator \(\left[l_{y}\left[l_{y}, l_{z}\right]\right]\) given that \(\left[l_{x}, l_{y}\right]=i \hbar l,\left[l_{y}, l_{z}\right]=i \hbar l_{x},\) and \(\left[l_{z}, l_{x}\right]=i \hbar l_{y}\)

The ground-state wavefunction of a hydrogen atom has the form \(\psi(r)=N \mathrm{e}^{-b r}, b\) being a collection of fundamental constants with the magnitude \(1 / a_{0},\) with \(a_{0}=53 \mathrm{pm}\) Normalize this spherically symmetrical function. Hint. The volume element is \(\mathrm{d} \tau=\sin \theta \mathrm{d} \theta \mathrm{d} \varphi r^{2} \mathrm{d} r,\) with \(0 \leq \theta \leq \pi\) \(0 \leq \varphi \leq 2 \pi,\) and \(0 \leq r<\infty,\) 'Normalize' always means 'normalize to 1 ' in this text.

Construct quantum mechanical operators in the position representation for the following observables: (a) kinetic energy in one and in three dimensions, (b) the inverse separation, \(1 / x,\) (c) electric dipole moment \(\left(\Sigma_{i} Q_{i} r_{i} \text { where } r_{i}\right.\) is the position of a charge \(Q_{i}\) ), (d) \(z\) -component of angular momentum \(\left(x p_{y}-y p_{x}\right),\) (e) the mean square deviations of the position and momentum of a particle from the mean values.

Confirm that the operators (a) \(T=-\left(\hbar^{2} / 2 m\right)\left(d^{2} / d x^{2}\right)\) and (b) \(l_{z}=(\hbar / \mathrm{i})(\mathrm{d} / \mathrm{d} \varphi)\) are Hermitian. Hint. Consider the integrals \(\int_{0}^{L} \psi_{a}^{*} T \psi_{b} \mathrm{d} x\) and \(\int_{0}^{2 \pi} \psi_{a}^{*} l_{z} \psi_{b} \mathrm{d} \varphi\) and integrate by parts.
See all solutions

Recommended explanations on Chemistry Textbooks

Physical Chemistry

Read Explanation

Chemistry Branches

Read Explanation

Inorganic Chemistry

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Nuclear Chemistry

Read Explanation

Kinetics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free