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 Partial derivative https://www.youtube.com/watch?v=kdMep5GUOBw https://www.youtube.com/watch?v=dfvnCHqzK54 finding partial derivative = finding a slope partial derivative with respect to x is like below and the partial derivative with respect to y is like below Confirm partial derivative by graphs the partial derivative with respect to x  the partial derivative with respect to y The formal definition of partial derivatives The symmetry of second partial derivatives Gradient  https://www.youtube.com/watch?v=tIpKfDc295M ∇ = is being a vector full of partial derivative operators Gradient and graphs https://www.youtube.com/watch?v=_-02ze7tf08 Gradient ∇f is shown like this on the graph. * which means the direction of the steepest ascent. Directional derivative Divergence formula  only x component gradient value. only y component gradient value Laplacian intuition f(x,y) Δf(x,y)  div(grad f) = ∇ㆍ∇f what does the divergence represent?  gradient / direction to in...

________________________________________________________________________________________________ MIT Gilbert Strang : Lecture 14 ( Orthogonal Vectors and Subspaces ) https://www.youtube.com/watch?v=YzZUIYRCE38&list=PL49CF3715CB9EF31D&index=14 Orthogonal vectors & Subspaces nullspace ⊥ row space N(A^T * A) = N(A)  * The angle between the two sub-spaces is 90 degrees (Orthogonal) Orthogonal vectors Pythagoras figured out that vectors x and y are orthogonal when x^T * y = 0 * So, 2 x^T * y = 0, when they are orthogonal ** Subspace S is orthogonal to subspace T means : every vector in S is orthogonal to every vector in T * for the case of orthogonal, they don't intersect in any non-zero vector. row space is orthogonal to nullspace. why? Ax = 0, so dot products of (row of A) · x = 0 which means the null space of x is orthogonal to every row. null space and row space are orthogonal complements in R^n Null space contains all vectors ⊥ row space. Comming : Ax = b   wh...