### Why parallelogram area is |ad-bc|?

Here is my question.

The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students.

### 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 null spa

### 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

### left-associative and right-associative

Japanese version I forget to explain one rule that function application is ``left-associative.'' f x y = (f x) y, The function is processed from left to right. For example, 1 - 2 - 3 is not a λ expression, but it is a left-associative example , means (1 - 2) - 3. Enclosed by parentheses '()' is calculated first. Therefore, 1-2-3 = (1-2)-3 = -1-3 = -4. If this is right-associative, 1-2-3 = 1-(2-3) = 1-(-1) = 2. Please note the answer is different. If we write this function as a λ expression, (λx. λy. λz. x - y - z) 1 2 3 This concludes (λ x. λy. λz. x - y - z) 1 2 3 ... apply x to 1 = (λ y . λz. 1 - y - z) 2 3 ... apply y to 2 = (λ z. 1 - 2 - z) 3 = (λ z . -1 - z) 3 ... apply z to 3 = -1 - 3 = -4. x = 1, y = 2, and z = 3 if this is left-associative and x = 3, y = 2, and z = 1 if this is right-associative. We use left-associative in λ calculus unless it is explicitly mentioned.