Computer Science/Linear Algebra 썸네일형 리스트형 12 Orthogonal Projections Projections onto subspaces Then we can say that like this. We already have seen that vector x in R can be represented by a vector in V and a vector in complement orthogonal. Visualizing a projection onto a plane The projection of x onto L, x - $proj_L x$ is orthogonal to L!! Projection is closted vector in subspace Projection of x onto v is the closest vector in our subspace to x. It's closer th.. 더보기 11 Orthogonal complements 11. Orthogonal complementsOrthogonal complementsV perp is equal, to the set of all x's, all the vectors x that are a member of our $R^n$, such that x dot V is equal to zero for every vector V that is a member of our subspace.Then what does this imply?What is a fact that a and b are members of V prep?√ That means a dot V, where this V is any member of our original subspace is V, is equal to 0 for.. 더보기 10 Transpose of a matrix Transpose of a matrix Let's define a transpose matrix. Transpose of a transpose matrix A is same with the matrix A! Determinant of transpose Let's assume determinant of any nxn matrix B is equal to the determinant of B's transpose. Then will it be the same if the matrix is (n+1) x (n+1)? Transpose of sums and inverses ) This means that C transpose is equal to the transpose of (A+B)! Rank(a) = ra.. 더보기 09 Linear transformation examples Linear transformation examples: Scaling and reflections Linear transformation examples: Rotations in R2 The rotation of some vector x is going to be equal to a counterclockwise theta degree rotation of x. Rotation in R3 around the x-axis You can generalize in R3. You can try this also in y, z axis. Unit vector unit vector is a vector that has length of 1. At any of dimension, it has length of 1. 더보기 08 Functions and linear transformations A more formal understanding of functions Range: subset of the codomain that the function actually maps to Vector transformation Linear transformations Let's see this Transformation is a lienar treanformation. We have to check the conditions. This conficts with the requirement for a linear tranformation. So this is not a linear transformation! Matrix vector products as linear transformations By m.. 더보기 07 Null space and column space Matrix vector products What we want to do now in this video is relate our notion of a matrix to everything we already know about vectors. Let's define what is means when we take the product of our matrix A with some vector x. Our definition will only work if the vector x has the same number of components as A has columns. This matrix A also can be represented like this. Now the interesting here .. 더보기 06 Matrices for solving systems Matrices: Reduced Row Echelon Form 1 The matrices are just arrays of numbers that are shorthand for this system of equations. We can make a coefficient matrix! Let's see this matrix is independent or not. We will transform this matrix to Reduced Row Echelon form(RREF (A)). The variables that you associate with your pivot entries, we call these pivot variables(x1, x3). And the variables that are .. 더보기 05. Vector dot and cross products Vector dot product and vector length link For multiplying vectors or taking the product, there's actually two ways. One of is the dot product. You signfy the dot product by saying a dot b. They borrowed one of the types of multiplication. This is not a vector, this will be a real scalar. So the dot product, you multiply two vectors and you end up with a scalar value. And let's define the length .. 더보기 이전 1 2 다음
