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5. (6 points each) Determine whether or not Wis a subspace of R", for each of...
Determine whether the set w is a subspace of R3 with the standard operations. If not, state why. (Select all that apply.) W = {(x1, 1/X1, X3): X1 and xy are real numbers, X1 + 0) W is a subspace of R W is not a subspace of R because it is not closed under addition. Wis not a subspace of R because it is not closed under scalar multiplication. X
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)...
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...
Determine whether the subset S is a subspace of R" or not. If it is a subspace, explain why it is, either by checking that the three defining properties of a subspace are satisfied or by using a result from class (for insta that the span of vectors subspace which is not satisfied (e.g. specific vectors u and v are in S but iu ö is not in S), Studying examples 3.38, 3.39 and 3.40 in the textbook could be...
6. For the following vector spaces V, determine if the subset H is a subspace. If not, give one reason why H fails to be a subspace. (a) (5 points) V is the set of functions f from R + R, and H is the set of polynomials of integer coefficients. (b) (5 points) V = P, is the vector space of polynomials of degree at most 2, and H is the subset of all polynomials in P2 of the...
Determine whether each of the following is a subspace of the relevant R". (a) V1 = {(x, y, z) | x, y ER, Z E Z} (b) V2 = {(2,4,4) + s(1, 2, 2) + t(4,5,7) | ste R} (c) V3 = {(a, b, c, d) | a, b, c, d e R, ab = 0}
3. For each of the following sets, determine if it is a subspace of R3. If it is a subspace, prove it. If is is not a subspace provide an example showing how it violates at least one of the subspace axioms (a) B , y,z) E R3 (x, y, 2)l 1) (b) S (a b, 3b+ 2a,a-b) a, be R) [10 (c) P (7,5,8) s(1,-1,2)t (3, 1,4) s,te R)
5) Given a subset S - lyx+z- -2(. Determine whether or not S is a subspace
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...
= 5. Determine if the following are linearly independent subsets: a) Determine whether or not vectors (1,-1,1,1), (3,0,1,1), (7,-1,2,1) form a linearly independent subset of R4. [1 01 To 27 -2 1] Let A= and C = . Do A, B, and C form 2 -1 -1 1 a linearly independent subset of M2x2? c) Determine if 5,x? – 6x,(3 – x)² form a linearly independent subset of F(-00,00). 6. Are the following bases? Why or why not. a) {(1,0,2),...