Problem 2. Let b, a₁,..., an be elements of a vector space, and as-sume that b € span{a₁,..., an}. Show that the sequence (A₁,..., An) in theexpressionb = λαι ++ Ananis unique if only if the vectors a₁, ..., an are linearly independent.

Answered: Image /qna-images/answer/e5f9cc39-406a-4e6d-870c-e48ce27894af.jpg

Answered: Problem 2. Let b, a₁,..., an be…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2.10 If A, B, and C are noncoplanar vectors (that is, A, B, and C are linearly independent), determine if the following set of vectors is linearly independent: r₁ = A- 3B +2C, r₂ = 2A-5B +3C, r3 = A - 5B + 4C.
Which of the following vectors are inearly independent? " B4 A) [1 2 3] [4 5 6] [7 8 9] B) [1 2 3] (2 4 6] (3 4 5) C) [2 2 -1] (4 0 2) [0 6 -1] D) [1 3 -2] [4 7 1] (3 -1 12] E) [1 2 1] [9 5 2] (7 1 0] F) [3 5 2] (1 7 6] [2 5 3] G) [2 5 2] [1 6 6] [4 H) (-9 5 21 (1 -5 6J (2 5 10 4] OG OA OE
Find the vectors T, N, and B at the given point. 16 Pus- (2. ²0₁0). (₁-15-2) r(1) = t), 3 3 T = N = B || K
3. Suppose A = 3 1 1 -2 (a) Find a vector X E R² such that AX = (b) Find a vector X € R² such that AX = B 9
(1) Recalled that for ₁, U2, U3, U₁ € R¹, then the span of these four vectors are defined as Suppose span {0₁, 0₂, 03, 0₁} = {av₁ + bû₂ + cũ3 + du: a, b, c, d = R}. --0---0 = = are in the set called 2 3 7 V3 Determine whether the following vectors in Rª (). = (O) 3 and ₁= () span {1, 02, 03, 04₁}. Hint: You should make an argument that minimize the amount of computation, and don't have to solve some equations 5 times. Does the given set span R¹?
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||
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.
Which of the following vectors are linearly independent? A) [1 2 3] [4 5 6] [7 8 9] B) [2 2 -1] [4 0 2] [0 6 -1] C) [1 2 3] [2 4 6] [3 4 5] D) [1 3 -2] [4 7 1] [3 -1 12] E) [1 2 1] [9 5 2] F) [3 5 2] [1 7 6] [2 5 31 G) [2 5 2] [1 6 6] [4 H) [-9 5 2] [1 -5 6] 12 5 -91 17 1 ol 10 41
Could a set of three vectors in R* span all of R*? Explain. What about n vectors in R™ when n is less than m? Could a set of three vectors in R* span all of R*? Explain. Choose the correct answer below. A. No. There is no way for any number of vectors in R* to span all of R. B. Yes. Any number of vectors in R* will span all of Rª Yes. A set of n vectors in R™ can span R™ when n
()- If then C(A) contains all vectors b2 by such bs by that O a. b 2b, and b, 3b, O b.by b ba Oc 2b b and 3by b d. by 2b, and b y 3b 23
If 1 and 2 are vectors in R³ and 7₁ 72, then 1 and 2 must be linearly independent. True O False

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Problem 2. Let b, a₁,..., an be elements of a vector space, and as- sume that b € span{a₁,..., an}. Show that the sequence (A₁,..., An) in the expression b = λια1 + + Anan is unique if only if the vectors a₁, ..., an are linearly independent.