1. Prove that each of the following is a subspace. (a) W = {x: x =...
0/1 pts Inooreat Question 9 Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why WR Two subspaces are the same when one subspace is a subset of the other subspace. Two subspaces are the same when they are spanned by the same vectors Two subspaces are the same when they are subsets of the same space Two subspaces are the same when they have the same dimension Incorrect 0/1 pts Question 10 Let U...
Let V = M2x2 be the vector space of 2 x 2 matrices with real number entries, usual addition and scalar multiplication. Which of the following subsets form a subspace of V? The subset of upper triangular matrices. The subset of all matrices 0b The subset of invertible matrices. The subset of symmetric matrices. Question 6 The set S = {V1, V2,v;} where vi = (-1,1,1), v2 = (1,-1,1), V3 = (1,1,-1) is a basis for R3. The vector w...
For the following six questions, indicate whether the following statements are true or false. In each case give a reason for your answer. Problem 13 [10 pts) If L:V +W is a linear transformation of vector spaces and U CW is a subspace of W, then {v € V | L(v € U} CW is a subspace of V. Problem 14 (10 pts) The set {A E R2x2 | A is nonsingular} is a subspace of R2x2. Problem 15 (10...
part a and b PROBLEM (HAND-IN ASSIGNMENT) Use the Subspace Test to determine whether the following sets W are subspaces of the given vector spaces: (A) The set W to be of all triples of real numbers (x, y, z) satisfying that 2x - 3y + 5z = 0 with the standard operations on Ris a subspace of R3. (B) The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication.
Please answer questions 2&3. Thank you! Remember that: A subspace is never empty, and is either the just the zero vector. i.e. [0), or has an infinite number of vectors A basis for a subspace is a set of t vectors. where t is the dimension of the subspace (usually a small number.) These vectors span the subspace and are linearly independent. This means that 0 can never part of a basis. The basis of the subspace (0) is empty....
4. (25 points) Which of the following subsets of R3 are subspaces.Explain. a) {(x, y, z) 1 x 0, y 0, z ? c) {(z, y, z) | x2 + y2 + z2-1} d) Is the set H of all matrices of the form |(a,0)T, (b,d)T] a subspace of the space of all 2x2 matrices with the usual matrix addition and scalar multiplica- tion?
Prove that the set W = {(x, y, z) * + = 0} is a subspace of Rs and then find a basis in W.
Hi, could you post solutions to the following questions. Thanks. 2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
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...
1. Use Cramer's rule to solve this system. X + 4z = 2 2x + y -z= 1 X +z=-1 on 910 2. Given the data points (0,), (1,3), (2,5) use the equation y=f(x) = mx +b to find the least square solution for best line fit. a. Evaluate the equation using the data points to obtain four linear equations. b. Write the system in 10a in a matrix form Ax=b. durants bhonebnih 516 10 boer word c. Write the...