15. Projections onto Subspaces

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Recommends Aravindh Vasu · 2020-01 · Mathematics Stack Exchange · on “Transpose of a matrix and the product $A A^\top$”

I've been following 3Blue1Brown's Essence of Linear Algebra [LINK] , basically the question (1) is what is the geometric meaning of the transpose? I've watched Chapter 9-Dot Products and duality [LINK] . I can see that the transpose of a matrix has something to do with duality and dual spaces, but I'm not able to put my finger on it. There is an answer , 3B1B Transpose [LINK] , in this answer there is a sentence, When you transpose a matrix, you are actually making use of this vector-dual vector identification to change your transformation to act on the dual vectors instead of the original vectors. I've been reading this sentence again and again, but I'm having a hard time understanding the

Cites Pedro · 2015-03 · Mathematics Stack Exchange · on “How to divide a $1 \times n$ rowvector by a $n \times 1$ vector and the result of such a divsion”

Dividing by a vector is something that is undefined. Let me give you some intuition about matrices to know why. Generally a matrix can be seen as an operator. Roughly speaking a matrix is something that acts on vectors and gives back an other vector. You have row vector (aka 1xn matrix) and column vectors (aka nx1 matrix). If you use a row vector then you write $xA = y$, and then we say that matrix A manipulated row vector x to produce row vector y. If you use column vectors you write $Ax = y$, and we say that we manipulated x to produce vector y. Notice that with column vectors, we multiply from the right. In most mathematical and engineering fields people use column vectors and we multiply

Cites Nimitz14 · 2015-11 · Mathematics Stack Exchange · on “Confused regarding the interpretation of A in the least squares formula A^T A = A^T b”

So I'm was watching gilbert strang's lecture to refresh my memory on least squares, and there's something that's confusing me (timestamp included). [this video] In the 2D case he has $A=[1,1;1,2;1,3]$. In the lecture he talks about how A is a subspace in our n dimesional (in this case n=2) space. If you look at the top left point of the chalk board in that time stamp you will see he has written down $A=[a_{1},a_{2}]$. Here he was talking about the 3d case, and the $a$s were vectors that spanned a plane onto which we wanted to project. My problem is, I'm not quite sure how I'm supposed to understand the values for the 2D. Clearly the 2nd column is the x values, but can one say they span a space? T

Cites YohanRoth · 2017-04 · Mathematics Stack Exchange · on “Cannot understand part of the lecture on linear algebra”

I cannot understand how projecting vector $b$ onto vector $a$ gives us a side $e$ of a right triangle which we can express as $e = b - p$? Video is here [this video] [LINK]

Cites Swakshar Deb · 2020-05 · Mathematics Stack Exchange · on “How can $a^T(b-xa) = 0 $ means $xa^{T}a = a^Tb$?”

I was watching gilbert strang lectures on the projection matrix [this video] , at 5.17 he wrote that: $a^T(b-xa) = 0 $ That means: $xa^{T}a = a^Tb$ how can we write this expression? In my opinion isn't it should be: $ a^Txa = a^Tb $ ? Actually I am not good at linear algebra, so I know the question may be silly.

Cites Rotkiv · 2020-12 · Mathematics Stack Exchange · on “how can I show that for projection matrix $P^2 = P$ without applying trigonometry”

I am following MIT 18.06 lectures [this video] on linear algebra and would like to understand how one can show that $P^2 = P$ for a projection matrix. I understand that for 2-d case: $P = \frac{a a^t}{a^t a}$ $P^2 = \frac{a a^t}{a^t a} \frac{a a^t}{a^t a}$ I assume I should easily show it by using some properties of transpose matrices/vectors, but I couldn't quite figure out how to derive it. (I am familiar with the proof that relies on trigonometric formulas)