(a) is a subspace, since any line passing through origin is a subspace of R3 of dimension 1.
(b) this plane do not passes through origin.
But, any subset which does not contain 0 cannot be a subspace.
Hence, given plane is not a subspace
I. Determine whether each of the subsets below are subspaces of R. (a) The line through...
I. Determine whether each of the subsets below are subspaces of R. (a) The line through (2,-5,3) and the origin. (b) The plane parallel to the z, y plane two units above the origin.
1. Determine whether each of the subsets below are subspaces of R3. (a) The line through (2,-5,3) and the origin. (b) The plane parallel to the x, y plane two units above the origin.
Determine whether each of the subsets below are subspaces of R3. (a) The line through (2,-5, 3) and the origin. (b) The plane parallel to the x, y plane two units above the origin.
Determine whether each of the subsets below are subspaces of R3. (a) The line through (2,-5, 3) and the origin. (b) The plane parallel to the x, y plane two units above the origin.
Name: Math 23 6. (14 points) Determine whether the following subsets are subspaces of the given veeto r space. Either prove that the set is a subspace or prove that it is not (a) The subset T C Ps of polynomials of degree less than or equal to 3 that are of the form p(x)-1+iz+o2+caz3, where c,02, c3 are scalars in R. (b) The set s-a a,bERM22, that is, the subset of all 2 x 2 matrices A where a11-a22...
QUESTION 2. (a) Decide whether each of the following subsets of R’ is a subspace. Either provide a proof showing the set is a subspace of R3, or provide a counterexample showing it is not a subspace: [9 marks] (i) S= {(x, y, z) ER3 : 4.0 + 9y + 8z = 0} (ii) S = {(x, y, z) E R3 : xy = 0} (b) Determine for which values of b ER, the set S = {(x, y, z)...
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace of R3. Justify your answer. (b) For the subsets which are subspaces, find a basis and the dimension for each of them
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace...
For each of the objects described below, state whether it is parallel, orthogonal or neither to the parametric line I = 3t - 7 y = - +4 z = 2t +1 (a) the vector ū = [14, 8, -1] Parallel Orthogonal Neither (b) the plane 3x - y - 5z = 17 Parallel Orthogonal Neither (c) the line with vector equation [x, y, z) = (1,5, 1] + t[ 6,2, 4 Neither Orthogonal Parallel [1,0,0] + t1,5,1 + $5,...
(4) (a) Determine the standard matrix A for the rotation r of R
3 around the z-axis through the angle π/3 counterclockwise. Hint:
Use the matrix for the rotation around the origin in R 2 for the
xy-plane. (b) Consider the rotation s of R 3 around the line
spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a
basis of R 3 for which the matrix [s]B,B is equal to A from (a).
(c) Give...
(a). Find the equation of the plane through Po = (1,2,1) with normal vector i = (3,1,2) (b). Find the equation of a plane through Po = (2,3,1) and parallel to the plane P:3x + 2y -- z = 4 | Q4. Consider the line z-3 y-2 3 L, : * - - - L2: **** 2+5 y-3 -1 2 (i). Write the equations of both lines in parametric form (ii). Find the direction vectors V1, V2 of the lines...
Problem 6-20 points. This question is about vector spaces and subspaces. (a) Define the terms "vector space" and "subspace" as precisely as you can. (b) Consider a line through the origin in R2, for example, the r-axis. Explain why this line is, or is not, a subspace of R2 in terms of your definitions in (a). (c) Consider the union of two lines through the origin in R2, for example, the z- and y-axes. Explain why this union of lines...