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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1. Let S = {(1,0, 1), (0, 1, 0)} be two vectors in R3. Write the following vectors as a linear combination of the vectors in S if possible. If it's not possible, show the matrix indicating why it isn't. (a) (-4,6, –4) (b) (-4,3, 3) (c) (0,24,0) (d) (1,0,0)
Please solve quckily
5. For which real values of 1 do the following vectors form a linearly independent set in R³?
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?
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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
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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.