3. Given pairwise orthogonal vectors u, v, w ER(each vector is orthogonal to every other), with...
Let w be a subspace of R", and let wt be the set of all vectors orthogonal to W. Show that wt is a subspace of R" using the following steps. a. Take z in wt, and let u represent any element of W. Then zu u = 0. Take any scalar c and show that cz is orthogonal to u. (Since u was an arbitrary element of W, this will show that cz is in wt.) b. Take z,...
3 - 2 Let u= Note that {u, v, w} is an orthogonal set of vectors and w - -3 4 9 be a vector in subspace W, where W = Span{u, v, w}. Let y= 11 -27 Write y as a linear combination of u, v, and uw, i.e. y = ciu + cqũ + c3W. Answer: y=
Given the following vectors u and v, find a vector w in R4 so that {u, v, w} is linearly independent and a non- zero vector z in R4 so that {u, v, z} is linearly dependent: 1-3 8 -8 -2 u = V= 5 -4 10 0 w=0 1- z=0 0
Lin Alg. vector in Span(u,u..,u,) to each of the vectors u. w...u. Show that w is orthogonal to every
Question 6) (9 points) Prove each of the following statements. (a) Suppose that the vectors {v, w, u} are linearly independent vectors in some vector space V. Prove then that the vectors {v + w, w + u,v + u} are also linearly independent in V. (b) Suppose T is a linear transformation, T: P10(R) → M3(R) Prove that T cannot be 1-to-1 (c) Prove that in ANY inner product that if u and w are unit vectors (ie ||vl|...
7. The set {u, v, w} is an orthogonal set of vectors, where u= (0,3,4), v = (1,0,0) and w = (0,4, -3). If (0,-1,-1) = au + bu + cw, then (a, b, c) = mark (x) the correct answer: A (-3,0,-) B (-2, 0, - 2) C (7,0, ) D(-2,0, 35) E (-7,0, -1) F (0,-1, -1)
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...
41 and w be vectors, and 39-42 Properties of Vectors Let u, V, and w be ved let c be a scalar. Prove the given property. 39. u. v = v.u 40. (cu) v = c(u.v) = u • (cv) 41. (u + v). w = uw + v.w 42. (u - v)•(u + v) = | u |2 - 1 v 12
Problem 9. Determine if the following pair of vectors are orthogonal. -3 13 -3 0 -7'0 25 -22.5 Problem 10. Prove the parallelogram law: where u and are vectors in IR Problem 11. Suppose a vector r is orthogonal to both vectors y and z. Prove that r is orthogonal to any vector in spany,
3. Consider two vectors u = 2i -j +2k and v=3i+2j-k. (a) Find a vector orthogonal to a and b. _ [3 marks] (b) Show that the vector from (a) is orthogonal to a and b. [1 mark]