Exploring the Role of Determinants in Orienting Scientific Research

[Lambda96]

Homework Statement

Show that the vector $W_n$ is orthogonal to all vectors $w_j$

Relevant Equations

none

Hi,

unfortunately, I have problems with the task c

I used the tip with the Laplace evolution theorem and rewrote the determinant to calculate $W_n$. Then I simply formed the scalar product $W_1 W_n$ and here I get now no further.

 

[Lambda96]

I hope you can read my calculation well, otherwise I will upload the calculation again in a higher resolution

 

[pasmith]

To get to this point, you expanded a determinant by its final column. What happens if you reverse that process after calculating the dot product?

Hint: Write $\mathbf{w}_n = D_1\mathbf{e}_1 + \dots + D_{n}\mathbf{e}_{n}$.

 

[Lambda96]

Thanks pasmith for your help and the tip

I have now calculated the scalar product with the $\vec{W_1}$, so $\vec{W_1} \cdot \vec{W_n}$ Now if I reverse the process again, that is the Laplace expansion then the unit vectors $\vec{e_i}$ disappear due to the scalar product and the column with the unit vectors in the determinant is replaced by the vector $\vec{W_1}$, so I get the following determinant.

$$det \left( \begin{array}{rrrrr}
W_1^1 & W_2^1 & \cdots & W_{n-1}^1 & W_1^1 \\
W_1^2 & W_2^2 & \cdots & W_{n-1}^2 & W_1^2 \\
\vdots & \vdots & \cdots & \vdots & \vdots \\
W_1^n & W_2^n & \cdots & W_{n-1}^n & W_1^n\\
\end{array}\right)$$

The first column now occurs twice, making the determinant zero.

 

[malawi_glenn]

This is a typo, right?

should be $w_1^2$

 

[Lambda96]

You are right malawi_glenn, it is a spelling mistake, it should be $W_1^2$, thanks for pointing it out, I will tell my lecturer so he can correct it