6. (20 points) Let W be a plane spanned by the vectors ői = [1, 2,...
#8. Let W be the subspace of R3 spanned by the two linearly independent vectors v1 = (-1,2,2) and v2 = (3, -3,0). (a) Use the Gram-Schmidt orthogonalization process to find an orthonormal basis for W. (b) Use part (a) to find the matrix M of the orthogonal projection P: R W . (c) Given that im(P) = W, what is rank(M)?
Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orthonormal set using the dot product on R. (Hence it forms an orthonormal basis for W.) (b) Let w = (-1,1,5,5) EW. Using the formula in the box above, express was a linear combination of u and u. (c) Let v = (-1,1,3,5) = R'. Find the orthogonal projection of v onto W.
(12 points) Let vi = 1 and let W be the subspace of R* spanned by V, and v. (a) Convert (V. 2) into an ohonormal basis of W NOTE: If your answer involves square roots, leave them unevaluated. Basis = { (b) Find the projection of b = onto W (c) Find two linearly independent vectors in R* perpendicular to W. Vectors = 1
11 -14 (1 point) Let W be the subspace of R3 spanned by the vectors 1 and 4 Find the projection matrix P that projects vectors in R3 onto W
Find an orthonormal basis for the subspace of R3
spanned by
Extend the basis you found to an orthonormal basis for R 3 (by
adding a new vector or vectors). Is there a unique way to extend
the basis you found to an orthonormal basis of R3 ?
Explain.
Let W be the subspace spanned by the given vectors. Find a basis for Wt, 0 1 A. W1 = W2 = 3 2 -1 2 B. W W2 2 -3 W3 = 6
(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-
Find a basis for the vector space W spanned by the vectors$$ \overrightarrow{v_{1}}=(1,2,3,1,2), \overrightarrow{v_{2}}=(-1,1,4,5,-3), \overrightarrow{v_{3}}=(2,4,6,2,4), \overrightarrow{v_{4}}=(0,0,0,1,2) $$(Hint: You can regard W as a row space of an appropriate matrix.)Using the Gram-Schmidt process find the orthonormal basis of the vector space W from the previous questionLet \(\vec{u}=(2,3,4,5,7)\). Find pro \(j_{W} \vec{u}\) where \(\mathrm{W}\) is the vector subspace form the previous two questions.
Let E be the plane in R3 spanned by the orthogonal vectors v1=(121)and v2=(−11−1) The reflection across E is the linear transformation R:R3→R3 defined by the formula R(x) = 2 projE(x)−x (a) Compute R(x) for x=(1260) (b) Find the eigenspace of R corresponding to the eigenvalue 1. That is, find the set of all vectors x for which R(x) =x. Justify your answer.
Let uj = [1,1,1]* and u2 = [1,2,2]t be vectors in R3 and V be the vector space spanned by {u1, U2}. a. 6pt Use Gram-Schmidt orthogonalization to find an orthonormal basis for V. b. 4pt Let w = [1,0,1)+. Find the vector in V that is closest to w.