Let u= -3 2 4 ; and let L denote the line thru the origin of R3 in the direction of u. The projection of R3 onto L — denoted PL : R3 −→ R3 — is definded to be equal to the projection pu onto the vector u. You may assume that PL is a linear transformation. Find the standard matrix [PL] for PL.
Problem 6. Let E be the plane: 2xi- x2 x3 = 0, and let P R3R3 be the orthogonal _ projection onto the plane E. Let v 1 (1) What are the image and kernel of P? What is the rank of P? Give a geometric descrip- tion, without relying (2) Give four different vectors e R3 such that Px Pv. (Again, solve geometrically and do not use the matrix of P.) (3) Find Pv (4) Find the reflection of...
Let L in R 3 be the line through the origin spanned by the vector v = 1 1 3 . Find the linear equations that define L, i.e., find a system of linear equations whose solutions are the points in L. (7) Give an example of a linear transformation from T : R 2 → R 3 with the following two properties: (a) T is not one-to-one, and (b) range(T) = ...
(3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A- (3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A-
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let P2 R2 R2 be the orthogonal projection onto the line L2: x32 2r2 0. 0. (1) What are the image and kernel of P2P What is the rank of P2P? Give a geometric description, without relying on the matrix of P2P (2) Find the matrix that represents P2P Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let...
Problem 24 : Let b b2 b3 ba E R4 be a fixed vector, b + 0. Define L:R4 R by C1 12 L(x) = b. I, 2= ER4. 13 24 where bºx is the dot product of b and x in R4. (a) Show that L is a linear transformation. (b) Find the standard matrix representation of L. (c) Find a basis for and the dimension of Ker L. (d) Is L one-to one? Explain why. (e) Is L...
Problem 6. (15 pts.) Project the vector b = (1, 2,5) onto the line spanned by the vector (2,3,4). Use the linear algebra viewpoint and notation, NOT the multi- dimensional calculus one. Show work to justify your answers to the following: (a) Find the projection vector p. (b) Find the projection matrix P. (c) Find the error vector e.
5. (3) Let X = Serie x = ()m - (*) indo 3 x 3 mais Find a 3 x 3 matrix A such that the projection PAX = e vector X ER projected onto the line I that is parallel to a and passes through the origin. Pex - Ax 6. (4) The line l in Ris given by the equation x + 3y = 0. (a) What is the angle e between the positive x-axis and the line...
Tbi b2 Problem 24 : Let b e R4 be a fixed vector, b+0. b3 b4 Define L:R4 → R by 11 12 L(x) = 6-2, x= ER 23 24 where b.x is the dot product of b and 2 in R4. (a) Show that L is a linear transformation. (b) Find the standard matrix representation of L. (c) Find a basis for and the dimension of Ker L. (d) Is L one-to one? Explain why. (e) Is L onto?...
Let L in R 3 be the line through the origin spanned by the vector v = 1 1 3 . Find the linear equations that define L, i.e., find a system of linear equations whose solutions are the points in L. lil (6) Let L in R3 be the line through the origin spanned by the vector v= 1. Find the linear equations that define L, i.e., find a system of linear equations whose solutions are...