Why is the normal transformation a inverse transpose? (4)

Normal vector as a perpendicular vector of the surface tangent vectors

Normal vector has the same direction to the cross product of two tangent vectors of a surface. Figure 2 shows the tangent vectors are correctly transformed by the matrix M that magnifies only x direction. However, their cross product is not necessary to the same as the transformation of normal by the matrix M.

Figure 2. The normal vector n is a cross product of tangent vectors u and v. Tangent vectors are linear to M, but not for the normal vector.

In short, tangent vectors u, v can be transformed by M, but their cross product is not. In general,

Are you convinced this is the reason distinguishing a usual vector and a normal vector? If you think about the x component of the cross product, uy vz - uz vy, this is not linear. Therefore, a linear transformation cannot transform this. This is my first explanation.

Comments

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix. Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume x where x is in the null space of A . The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the n...

My solution of Google drive hang up at "One moment please"

Today I installed Google drive to my Windows 7 environment to share files with my Linux machines. After sign in, the application window said "processing," then it hanged up. There was a button "you must enable javascript". I pushed it, then "One moment please..." after 5 minutes, I exited the program tried it again. It seems some security setting causes this problem. My solution: set https://accounts.google.com as a trusted site. Procedure: Open the control panel Go to network and control Go to Internet Options Open Security Tab Click Trusted sites Click the "site" button copy & paste https://accounts.google.com to "Add this website to the zone" and click Add button Now it worked for me. But if I removed this site, it still works. That puzzled me a bit...

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