bination of (v₁,..., vn). Then we must have m≤n, and there is a replacement of m of the vectors u, by (u1,..., um), such that after renaming some of the indices of the vis, the families (u1,..., Um, Um+1,..., Un) and (v₁,..., vn) generate the same subspace of E.

bination of (v₁,..., vn). Then we must have m≤n, and there is a replacement of m of the vectors u, by (u1,..., um), such that after renaming some of the indices of the vis, the families (u1,..., Um, Um+1,..., Un) and (v₁,..., vn) generate the same subspace of E.

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

Compute the norm of the vector 7 = |||
Suppose {v1, v2, ..., vp} is an orthogoal set of vectors. Show that {c1v1, c2v2, ..., cpvp} will also be an orthogonal set, for any choice of scalars c1, c2, ..., cp.
Suppose X = {-3,-2,-1, 0, 1, 2, 3}. Let fuzzy subsets A and B of X be defined by their membership vectors A = [0.0, 0.3, 0.8, 1.0, 0.8, 0.3, 0.0] and B = [1.0, 0.9, 0.7, 0.5, 0.2, 0.0, 0.0] Determine the membership vectors (write the results as list of numbers rounded to 2 decimal places) Operation Model ANB minimum ANB product ANB drastic product AUB maximum AUB probabilistic sum AUB drastic sum standard
6. Let 7₁, 7₂, .... that {V₁, V₂, . . . . . ., Ủn, } is linearly dependent. Un be vectors in a vector space V and let 7 be any vector in the span of S = 9 Ủi, U2,... 2 Un}. Prove
3. Prove the following to establish theorem 7. (a) If the set of vectors {₁, 2, 3, ...} in R" are linearly dependent then at least one of the vectors can be expressed as a linear combination of the others. (b) If at least one of the vectors from the set containing vectors ₁, 2, 3, ... in Rn can be expressed as a linear combination of the others then the vectors form a linearly dependent set.
S is a linearly independent set,then for any x element of , x is not the linear combination of other vectors in S. Is this true or false?
Suppose v1, v2, v3, 04, Ū5 and ve are vectors in R6 satisfying the following linear relation: 201 + 302 – Ủ4 + 205 - Which of the following must be true? (Select all that apply) span(v1, v2, v3, 4, 06) = span(71, v3, 74, 05, V6) span(v1, v2, v3, 74, 05) = span(v1, v2, v4, V5, 76) span(v1, v2, v3, v4, v6) = span(v1, 02, v3, v4, Ü5 , v6) span(v1, v2, V3, v6)= span(73, v4, V5 , V6) O span(v1, v2, v3, V4, 05) = span(72, 03, v4, 05, V6 )
1- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (-3,0,4), (5,-1,2), (1,1,3) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 2- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthonormal basis for R³, then obtain an orthonormal basis from them. Express (3,7,-4) as a linear combination of the orthonormal basis. 3- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (1,1,0), (0,0,1), (0,1,1) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 4- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthogonal basis for R³, then obtain an orthonormal basis from them. Express (2,3,-4) as a linear combination of the orthonormal basis. 5- Find the coordinates of the vector u = (0,25, 1) with respect to the orthogonal basis v₁ = ( 3,4,0), V₂ = (−4, 3, 0 ) and v₂ = (−4,3,0) 6- Find the coordinates of the vector U = (7,5,2) with respect to the basis V₁ =…