Q8. (a) Let y = [4, 8) and u = [3,1]. Write y as the sum of a vector in span{u} and a vector orthogonal to u. (b) Show that if U and V are n x n orthogonal matrices, then so is UV.
Let W be the subspace spanned by u, and up. Write y as the sum of a vector in W and a vector orthogonal to W. 2 y = 6 un 5 The sum is y=9+z, where y is in W and Z is orthogonal to W. (Simplify your answers.) N
6 Let y = and u Write y as the sum of two orthogonal vectors, one in Span (u) and one orthogonal to u. 5 7 y=y+z=( (Type an integer or simplified fraction for each matrix element. List the terms in the same order as they appear in the original list.)
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for all X, Y CR”, if X IY then Span(X) 1 Span(Y). (b) Let X and Y be linearly independent subsets of R”. Prove that if X IY, then X UY is linearly independent. (C) Prove that every maximally pairwise orthogonal set of vectors in R” has n + 1 elements. Definition: Let V be a vector space and let U and W be subspaces...
3 5 Let y = and us .Write y as the sum of two orthogonal vectors, one in Span {u} and one orthogonal to u. 8 -5 y=y+z=]] (Type an integer or simplified fraction for each matrix element. List the terms in the same order as they appear in the original list.)
Let U and V be nxn orthogonal matrices. Explain why UV is an orthogonal matrix. [That is, explain why UV is invertible and its inverse is (UV)'.] Why is UV invertible? O A. Since U and V are nxn matrices, each is invertible by the definition of invertible matrices. The product of two invertible matrices is also invertible. OB. UV is invertible because it is an orthogonal matrix, and all orthogonal matrices are invertible. O c. Since U and V...
Wite **the sum of two vectons, one in Span {u) and one in Span (wa). Assume that (.....) is an orthogonal besis Type an integer or simplified traction for each max element) Verity that {.uz) is an orthogonal sot, and then find the orthogonal projection of y onto Span(uz) y To verty that (0-uz) as an orthogonal set, find u, uz 2-0 (Simplify your answer.) The projection of yonte Span (0,2) 0 (Simplify your answers.) LetW be the subspace spanned...
#5
6.3.8 Let W be the subspace spanned by U, and up. Write y as the sum of a vector in W and a vector orthogonal to W. -1 -2 y = un = 3 2 -1 The sum is y = y +z, where y 8. is in W and z = Doo is orthogonal to W. (Simplify your answers.)
Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector orthogonal to W. 1 -1 6 u u2 6 1 1 4 1 y= (Type an integer or simplified fraction for each matrix element.)
Linear Algebra
-3 1. Let y=| 2 | and 1-1 I. (a) (8 pts) Find projuy (b) (4 pts) Find the component of y orthogonal to u. (c) (4 pts) Write y as the sum of a vector in Span u and a vector orthogonal to u (d) (4 pts) Find an orthogonal basis for R' which contains u.