hazbijav6

2022-10-08

If ${\alpha}_{ij}{A}^{i}{B}^{j}=0$ and ${A}^{i}$ and ${B}^{j}$ are arbitrary vectors, then prove that ${\alpha}_{ij}=0$

This problem appeared in my Differential Geometry class, the professor explained the problem by, first taking an arbitrary vector and demonstrating that ${\alpha}_{ii}=0$. and then proceeded to demonstrate that, ${A}^{l}={B}^{m}=1,(1\leqq l\le n,1\leqq m\leqq n,l\ne m)$. I get the proof somewhat. Can any of you elucidate it or give an alternative proof?

This problem appeared in my Differential Geometry class, the professor explained the problem by, first taking an arbitrary vector and demonstrating that ${\alpha}_{ii}=0$. and then proceeded to demonstrate that, ${A}^{l}={B}^{m}=1,(1\leqq l\le n,1\leqq m\leqq n,l\ne m)$. I get the proof somewhat. Can any of you elucidate it or give an alternative proof?

Johnathon Mcmillan

Beginner2022-10-09Added 7 answers

${A}^{i}{B}^{j}$ are the component of the tensor product of two vector $\mathbf{A}\otimes \mathbf{B}$. Among all these tensors there are also the tensors ${\mathbf{e}}_{i}\otimes {\mathbf{e}}_{j},$ where $B=\{{\mathbf{e}}_{1},\dots ,{\mathbf{e}}_{n}\}$ is a base of the vector space V,and ${B}^{\prime}=\{{\mathbf{e}}_{i}\otimes {\mathbf{e}}_{j},\text{}i,j=1,\dots ,n\}$ is a base of the space of rank 2 tensors over tensors over V.

${\alpha}_{i,j}{A}^{i}{B}^{j}$ is the inner product of the tensor $\mathit{\alpha}$ and the tensor $\mathbf{A}\otimes \mathbf{B}$

${\alpha}_{i,j}{A}^{i}{B}^{j}=\mathit{\alpha}:(\mathbf{A}\otimes \mathbf{B})$

You know that if the inner product of a vector for each element of a base vanish, then the vector is the null vector. This is true also for the inner product vector space of tensors.

${\alpha}_{i,j}{A}^{i}{B}^{j}$ is the inner product of the tensor $\mathit{\alpha}$ and the tensor $\mathbf{A}\otimes \mathbf{B}$

${\alpha}_{i,j}{A}^{i}{B}^{j}=\mathit{\alpha}:(\mathbf{A}\otimes \mathbf{B})$

You know that if the inner product of a vector for each element of a base vanish, then the vector is the null vector. This is true also for the inner product vector space of tensors.

rialsv

Beginner2022-10-10Added 3 answers

Fix, h,k, then if you take

$\begin{array}{r}{A}^{i}=\{\begin{array}{lll}1,& & i=h,\\ 0,& & i\ne h,\end{array}\end{array}\phantom{\rule{2em}{0ex}}\begin{array}{r}{B}^{j}=\{\begin{array}{lll}1,& & j=k,\\ 0,& & j\ne k\end{array}\end{array}$

then

${\alpha}_{ij}{A}^{i}{B}^{j}=0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{\alpha}_{hk}=0,$

for the arbitrariness of h,k, this is true for all h,k.

$\begin{array}{r}{A}^{i}=\{\begin{array}{lll}1,& & i=h,\\ 0,& & i\ne h,\end{array}\end{array}\phantom{\rule{2em}{0ex}}\begin{array}{r}{B}^{j}=\{\begin{array}{lll}1,& & j=k,\\ 0,& & j\ne k\end{array}\end{array}$

then

${\alpha}_{ij}{A}^{i}{B}^{j}=0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{\alpha}_{hk}=0,$

for the arbitrariness of h,k, this is true for all h,k.

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix

$$\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$$ Find, correct to the nearest degree, the three angles of the triangle with the given vertices

A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)Whether f is a function from Z to R if

?

a) $f\left(n\right)=\pm n$.

b) $f\left(n\right)=\sqrt{{n}^{2}+1}$.

c) $f\left(n\right)=\frac{1}{{n}^{2}-4}$.How to write the expression ${6}^{\frac{3}{2}}$ in radical form?

How to evaluate $\mathrm{sin}\left(\frac{-5\pi}{4}\right)$?

What is the derivative of ${\mathrm{cot}}^{2}x$ ?

How to verify the identity: $\frac{\mathrm{cos}\left(x\right)-\mathrm{cos}\left(y\right)}{\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)}+\frac{\mathrm{sin}\left(x\right)-\mathrm{sin}\left(y\right)}{\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)}=0$?

Find $\mathrm{tan}\left(22.{5}^{\circ}\right)$ using the half-angle formula.

How to find the exact values of $\mathrm{cos}22.5\xb0$ using the half-angle formula?

How to express the complex number in trigonometric form: 5-5i?

The solution set of $\mathrm{tan}\theta =3\mathrm{cot}\theta $ is

How to find the angle between the vector and $x-$axis?

Find the probability of getting 5 Mondays in the month of february in a leap year.

How to find the inflection points for the given function $f\left(x\right)={x}^{3}-3{x}^{2}+6x$?

How do I find the value of sec(3pi/4)?