October 2012, Malala Yousafzai has been shot by Taliban who against her campaign for children' rights of education. Here is subtitle of her full speech of her 16 years old birthday. English and Japanese. Please use a HTML5 and Javascript supported browser (Firefox, Chrome, IE10, ...) to watch the video. If you have a problem to watch the video, this page is another page to watch the video.
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...
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