Q 1 Let V C R3 be the subspace V = {(x,y, z) E R3 : 5x 2y z 0} a) Find a basis B for V. What is the dimension of V? b) Find a basis B' for R3 so that B C B'
Problem 3: Let f: X -> R, XC R2, be given by f(x, y)n(x 2y 1), V(r,y) e X Find the maximal domain X and write the second-order Taylor polynomial for f around the point (2,1) E X. (6 points)
Problem 3: Let f: X -> R, XC R2, be given by f(x, y)n(x 2y 1), V(r,y) e X Find the maximal domain X and write the second-order Taylor polynomial for f around the point (2,1) E X. (6 points)
4. Let = 0 , 4r + 2y+-2). M={(x,y,z) € R' | - Show that A/ is a one dimensional manifold and find the maximum and minimum values of SIM where f(x,y, z) = ry + z.
4. Let = 0 , 4r + 2y+-2). M={(x,y,z) € R' | - Show that A/ is a one dimensional manifold and find the maximum and minimum values of SIM where f(x,y, z) = ry + z.
Problem 5 (25 points). Let Mat2x2(R) be the vector space of 2 x 2 matrices with real entries. Recall that (1 0.0 1.000.00 "100'00' (1 001) is the standard basis of Mat2x2(R). Define a transformation T : Mat2x2(R) + R2 by the rule la-36 c+ 3d - (1) (5 points) Show that T is linear. (2) (5 points) Compute the matrix of T with respect to the standard basis in Mat2x2 (R) and R”. Show your work. An answer with...
*Let f : R2 -R be given by z, y)(0,0 r, y)- 2y and f(0,0) = 0. (a) Decide if both partial derivatives of f exist at (0, 0) (b) Decide if f has directional derivatives along all v R2 and if so compute these. (c) Decide if f is Fréchet differentiable at (0, 0)? (d) What can you infer about the continuity of the partial derivatives at (0, 0)? て
Let T є L(C3) be defined by T(r, y, z)-(y-2-2c, z-2-2y,1-2y-22). (a) Is span((1,1,1)) invariant under T? (b) Is U = { ( (c) Is U = {(x, y, z) : x + y + z = 0} invariant under T? (d) Is λ 2 an eigenvalue of T? Is T-21 injective? (e) Find all eigenvectors of T associated to the eigenvalue λ =-3. 4. r, y,r-y) : x, y E C} invariant under T?
How can I get the (a) 3*2 matrix A?
x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0 y (a) Find a 3x2 matrix A whose column space is V and the entries a a1 0 = (b) Find an orthonormal basis for V by applying the Gram-Schmidt procedure (c) Find the projection matrix P projecting onto the left nullspace (not the column space) of A (d) Find an SVD (A...
6. Let P be the subspace in R 3 defined by the plane x − 2y + z
= 0. (a) [5 points] Use the Gram–Schmidt process to find orthogonal
vectors that form a basis for P. (b) [5 points] Find the projection
p of b = (3, −6, 9) onto P.
6. Let P be the subspace in R3 defined by the plan 2y+z0 (a) [5 points] Use the Gram-Schmidt process to find orthogonal vectors that form a basis...
Let V be the set of vectors [2x − 3y, x + 2y, −y, 4x] with x, y R2. Addition and scalar multiplication are defined in the same way as on vectors. Prove that V is a vector space. Also, point out a basis of it.