Homework 5. (due Wednesday 8 th Nov midnight)

Transcription

1 Homework (due Wednesday 8 th Nov midnight) Use this definition for Column Space of a Matrix Column Space of a matrix A is the set ColA of all linear combinations of the columns of A. In other words, if A = [a, a,..,a n ] with columns in R m, then ColA is the same as span{ a, a,..,a n }..Find basis for each of the following subspaces: column space ColA c). Write a system of equations that is equivalent to the given vector equation.

2 . Write a vector equation that is equivalent to the given system of equations.. Check if the column vectors of matrix A are linearly independent or not.. Write the following vector equations as matrix equations.

3 . Write the augmented system for the linear system that corresponds to the matrix equation Ax = b. Solve the system and write the solution as a vector.. Describe all solution of Ax= in parametric vector form, where A is row-equivalent to the given matrix. 8. Let H be the set of all vectors of the form this show that H is a subspace of R (. t t t. Find a vector v in R ( such that H = Span{v}. Why does 9. Let W be the set of all vectors of the form b + c b c. Where b and c are arbitrary. Find vectors u and v such that W = Span{v, u}. Why does this show that W is a subspace of R (.. Given subspaces H and K of a vector space V, the sum of H and K, written as H + K, is the set of all vectors in V that can be written as the sum of two vectors, one in H and the other one in K, that is, H + K = {w: w = u + v for some u in H and some v in K} a. Show that H+K is a subspace of V. b. Show that H is a subspace of H+K and K is a subspace of H+K.

4 . Mark each statement True or False. Justify each answer. a. A subset H of a vector space V is a subspace of V if the zero vector is in H. b. A subspace is also a vector space. c. A vector space is also a subspace. d. R C is a subspace of R (.. Let W = span{v,. v p }. Show that if x is orthogonal to each v j, for j p, then x is orthogonal to every vector in W.. Let U = and x = U T U D p = p = = = = ku k Dkk. Notice that U had orthonormal columns and = p = = p = D =.. D D 9

5 kok. Mark each statement as True or False and justify your answer. In the exercise - determine which sets of vectors are orthonormal. If a set is only orthogonal, normalize the vectors to produce orthonormal sets.. Verify that {u, u } is an orthonormal set, and then find the orthogonal projection of y onto Span{u,u }. a. D D D b. R n S Df ;:::; p g i j D i j S m n A! A c c = = = : :8 :8 : = = = = = = p = p = p = p = p = p = = p = = = p = p = p = p 8 = p 8 = p 8 D D D k = = k

6 . D 8 = D = = W D f ; g D = = =. b. a. Let U = [u, u ]. Compute U T U and UU T. b. Compute proj W y and (UUT)y.. The given set basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. a. 9 9 D W R n f ;:::; p g f ;:::; q g W? f ;:::; p ; ;:::; q g R n W C W? D n a.. Find an orthogonal basis for the column space of each matrix below.

7 b Use the Gram-Schmidt process to produce an orthogonal basis for the column space of A D