Consider following three sets: (i) set A has p vectors of the vector space V, and A= span{V}, (ii) set B has q linearly independent vectors of V, (iii) set C has r vectors of V, and C is the basis of V. Which of the following inequality is true? (Note: p, q, and r are distinct positive integers). (a) p

Consider following three sets: (i) set A has p vectors of the vector space V, and A= span{V}, (ii) set B has q linearly independent vectors of V, (iii) set C has r vectors of V, and C is the basis of V. Which of the following inequality is true? (Note: p, q, and r are distinct positive integers). (a) p

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

3. Prove the following to establish theorem 7. (a) If the set of vectors {vi, 02, 03, ...) 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 vi, 02, 03, ...up in R" can be expressed as a linear combination of the others then the vectors form a linearly dependent set.
Suppose V₁, V₂, V3 is an orthogonal set of vectors in R 5. Let w be a vector in Span(V₁, V2, V3) such that V₁ • V₁ = 42, V₂ · V₂ = 131, V3 · V3 = 9, W.V₁ = -126, w · V₂ = -655, w · V3 = 27, then w = V₁ + V₂+ V3.
Suppose {V1, V2, V3} are linearly independent vectors in a vector space V. Find the number(s) k such that the vectors w₁ = V₁ + kv₂, W2 = V2 + kv3 and w3 = V3+ kv₁ are linearly dependent. Justify your answers. Hint: Assume C₁W₁+C₂W2+C3W3 0 for some scales C1, C2, C3. Express it in terms of lin- ear combinations of V₁, V2, V3. Then, use the linearly independence of {V₁, V2, V3} to get a linear system with k as a parameter and C₁, C2, C3 as unknown variables. {w₁, W2, W3} are linearly dependent if the system only have zero solutions.
Suppose that v = (v1, v2, ..., vn) and û = (u₁, U2, ..., Un) are a pair of n-dimensional vectors. Assume that each component of the vector is a real number, so 7 and u are both members of the set R¹. We will say that and u are "almost the same" when every component of is close to every component of ū. That is, v₁ is close to u₁, v2 is close to u2, etc (practically speaking, "close" means that their absolute difference is small). Assume that we are given the predefined predicate CloseTo(x, y) and the integer constant n. Use them to write a formal definition of the new predicate Almost The Same (7, u) which asserts that n dimensional vector is almost the same as ū. Tip: It is not legal to say i v to refer to a component of v, because is not a set. Instead, use vi to refer to the ith component of v. What set would i belong to in this case?
If {v1, v2, . . . , vn} is a linearly dependent set of vectors, prove thatone of these vectors is a linear combination of the other.
Let u1, U2, ..., un be a set of vectors, and suppose there are nontrivial solutions to x11x2u2+ +xnun = 0 Explain why, if x is a free variable, you can always express u; in terms of the vectors corresponding to the basic variables.
For any two nonparallel vectors a and b, it holds that Select one: O O O None of them 3a + b 3|a| + |b| O|3a + b = 3|a| + |b||
If V1, V2, V3, V4, and W are all vector spaces. Are L(V1x...xV4, W) and L(V1, W) x...x L(V4, W) isomorphic vector spaces?
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.
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 )
(b) In each part, determine whether the given vector – x² + x + 3 € P3(R) is in the span of S = {x³ + x² + x + 1, x² + x + 1, x + 1 }. ii. i. 2x3 -x³ + 2x2 + 3x + 3 € P3 (R) is in the span of S = {x³3 + x² + x + 1, x2 + x + 1, x + 1}.
8 a) 4 B) 3 NS) 2nd D) 0 TO) one