Fast lookup child

L is not a so diligent girl. But she sometimes finishes the exercise very quickly. Although I found she repeatedly mistook the same problem. For example, in one exercise, 4+7 was always 12. I asked what is equal to 4 + 7. She used fingers, took a little while, but she answered correctly at the end. Then I noticed, ``Wait, if she solve the problem with this speed, how she can finish the exercise so quickly?'' So, I sat down next to her and watched how she solved the problem.

Surprise! She doesn't calculate at all. She just looks up the last pages and copies the answers. What I was impressed was she was so fast. She was faster than me to look up the questions. I understood why she made the same mistakes again and again. I learned a new German sentence, ``Bitte nicht angucken!'' This means, ``Don't cheat by looking.'' though it's hard to translate to English.

Her speed of cheating is impressive, this might be her talent. I don't know that I shall discourage her talent(?). That's a tough call for me, but I believe it is better to know to calculate numbers. So, today, I said ``Bitte nicht angucken!'' to her again. But I still don't know. Teaching is a tough business.

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

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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...

Lx=d: SundayResearch

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