**Prove that two vectors are linearly dependent (taken as a set of vectors) if and only if one vector is a multiple of the other.**This statement explores the concept of linear dependence in vector spaces. If two vectors are linearly dependent, it means that they lie along the same line in a vector space, implying that one is a scalar multiple of the other. Essentially, neither vector adds a new dimension to the span of the set, leading to a dependency between them. Understanding this relationship is crucial in linear algebra, as it forms the basis for more complex analyses of vector spaces and their dimensions.

Let X and Y be two vectors. Suppose these two vectors are linearly dependent. [To prove: One vector…

Answered: Prove that two vectors are linearly…

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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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?
The set of vectors {(1,1,0,0),(0,-1,0,0),(0,0,1,1),(1,0,1,1)} is linearly dependent. What is a valid expression of the last vector as a linear combination of the others? a. (1,0,1,1)=(1,1,0,0)+(0,-1,0,0)-(0,0,1,1) b. (1,0,1,1)=(1,1,0,0)+(0,-1,0,0)+(0,0,1,1) c. (1,0,1,1)=-(1,1,0,0)+(0,-1,0,0)+(0,0,1,1) d. (1,0,1,1)=2(1,1,0,0)+2(0,-1,0,0)+2(0,0,1,1)
2. If a set of vectors contains at least 2 vectors that are scalar multiples of each other, then the set is linearly dependent.
4. ) Determine whether the following three vectors are linearly independent or inearly dependent. v₁ = (3,3, 0, 2), v₂ = (0, 2, 3, 1), V3 (3, 0, -2, -1). Hint: You can turn them into column vectors. =
There are 6 linearly independent vectors in R5. O True False
Let v,=( 2, 5, 3 ), v2 = ( 1, 1, 1 ) and v3 = (4 , -2, 0 ) are vectors in R3 then are .a Linear independent .b Linear dependent

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Prove that two vectors are linearly dependent (taken as a set of vectors) if and only if one vector is multiple of the other.