Here ê, are orthonormal unit base vectors in R³. 2.12 Let the vectors (i, j, k) constitute an orthonormal basis. In terms of this basis, define a cogredient basis by e₁=-Î-Ĵ, e₂ e2 = 1+2ĵ-2k, e3 e3 = 2î+j+ k. = Determine (a) the dual or reciprocal (contragredient) basis (e¹,e², e³) in terms of the orthonormal basis (î, j, k), (b) the magnitudes (or norms) |e₁|, |e2|, |e3|, |e¹|, |e²|, and |e³|, and (c) the cogredient components A1, A2, and A3 of a vector A if its contragredient components are given by A¹ = 1, A² = 2, A³ = 3.

Here ê, are orthonormal unit base vectors in R³. 2.12 Let the vectors (i, j, k) constitute an orthonormal basis. In terms of this basis, define a cogredient basis by e₁=-Î-Ĵ, e₂ e2 = 1+2ĵ-2k, e3 e3 = 2î+j+ k. = Determine (a) the dual or reciprocal (contragredient) basis (e¹,e², e³) in terms of the orthonormal basis (î, j, k), (b) the magnitudes (or norms) |e₁|, |e2|, |e3|, |e¹|, |e²|, and |e³|, and (c) the cogredient components A1, A2, and A3 of a vector A if its contragredient components are given by A¹ = 1, A² = 2, A³ = 3.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Pls ans all part
Find an orthonormal basis of the plane x12x2x3 = 0. 1 2 [廣告] ਦੇ Answer:
Let {1, 2, 3, 4, 5, 6} be the standard basis in R6. Find the length of the vector = 3₁ - 22 - 2ě3 - 2ě4 – 2e5 - 3ē6. #
) Consider the vectors -- (-) --- () --- () a = b= = Do these vectors form a basis of R³? If yes, prove that they do, if no explain why.
Show that the vectors u₁ = (3, 1,-4), = (3,1, –4), u₂ = (2,5,6), and u} = = (1, 4, 8) form a basis in R³. Find the coordinates of (1, 1, 1) in this basis.
(3,0,0), v2 = (1,2,0), and v3 = a) Are the vectors linearly independent or linearly dependent? Given vectors v, = (2, –4,5). b) Do the vectors form a basis for R3? Why or why not?
Exercise 4.3 Find all the vectors of R³ which have the same coordinates in the canonical basis and in the basis = {(5, 18, 22), (-1,-8,-1), (1,-1, 12)}.
Let A = 0 -2 0 2 -3 -2 2 1 -4 2 -2 1 6 4 -3 -2 4 -2 2 a. A basis for the column space of A is { }. You should be able to explain and justify your answer. Enter a coordinate vector, such as or , or a comma separated list of coordinate vectors, such as , or ,. b. The dimension of the column space of A is answer): A. rref(A) has a pivot in every row. B. The basis we found for the column space of A has three vectors. OC. Three of the five columns in rref(A) have pivots. OD. Two of the five columns in rref(A) have pivots. □ E. rref(A) has a pivot in every column. □ F. rref(A) is the identity matrix. c. The column space of A is a subspace of d. The geometry of the column space of A is the origin inside R^5 because (select all correct answers -- there may be more than one correct because each column vector in A is a vector in R^4
Find an orthonormal basis of the plane r + 7r2 - r3 = 0.