3, (10%) Let V be the subset of R3 consisting of vectors of the form (a,...
3, (10%) Let V be the subset of R3 consisting of vectors of the form (a, b, a). Determine whether V is a subspace of R3. If it is a subspace, give a basis and its dimension
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
(22). (10 Marks). Consider vectors v = 0 - in R3 (a). (5 marks) Find a basis of the subspace of R3 consisting of all vectors perpendicular to v. (b). (5 marks) Find an orthonormal basis of this subspace.
thank you very much Exercise 4.10.52 Consider the vectors of the form :11, v, w ER〉 2w-2v-811 Is this set of vectors a subspace of R3? If so, explain why, give a basis for the subspace and find its dimension.
(b) Let H be the subset of R3 consisting of all points that can be expressed 2 in the form y where x is any real number, and y is any real number 31 such that y > 0. Is H a subspace of R3? Be sure to support your claim! (5 points)
Let V = R3 be a vector space and let H be a subset of V defined as H = {(a, b,c) : a? = b2 = c}, then H Select one: O A. satisfies only first condition of subspace B. is a subspace of R3 C. None of them O D. satisfies second and third condition of subspace
1. Let az, az, az, a4 are vectors in R3. Suppose that az 3a1 – 2a3 + 84. (a) Are aj, aj, az, a4 linearly independent? (b) Suppose that ai, az, a4 are linearly independent. What is the dimension of the span{a1, az, az, a4}? (c) Is the set of vectors aj, az, az, a4 form a basis of R3? Explain your reasoning. (d) Form a basis of R3 using a subset of ai, a2, a3, 24.
Q 1 Let V C R3 be the subspace V = {(x,y, z) E R3 : 5x 2y z 0} a) Find a basis B for V. What is the dimension of V? b) Find a basis B' for R3 so that B C B'
Question (7) Consider the vector space R3 with the regular addition, and scalar aL multiplication. Is The set of all vectors of the form b, subspace of R3 Question (9) a) Let S- {2-x + 3x2, x + x, 1-2x2} be a subset of P2, Is s is abasis for P2? 2 1 3 0 uestion (6) Let A=12 1 a) Compute the determinant of the matrix A via reduction to triangular form. (perform elementary row operations) Question (7) Consider...
Problem 1: consider the set of vectors in R^3 of the form: Material on basis and dimension Problem 1: Consider the set of vectors in R' of the form < a-2b,b-a,5b> Prove that this set is a subspace of R' by showing closure under addition and scalar multiplication Find a basis for the subspace. Is the vector w-8,5,15> in the subspace? If so, express w as a linear combination of the basis vectors for the subspace. Give the dimension of...