Let V be a vector space and S a subset of V with the property that whenever v1, v2, . . . , vn ∈ S and a1v1 +a2v2 + · · · + anvn= 0, then a1 = a2=  · · = an = 0. Prove that every vector in the span of S can be uniquely written as a linear combination of vectors of S.

Q: Let W be the subset of the 3-dimensional vector space R° defined by W X = E R3 2x1x2 = x3 x3 Which…

A: We will see which vector will satisfies the condition of 2x1x2=x3.

Q: Suppose u1,..., u, and v1, . Span {u¡, ..., u„} and K = Show that H + K = Span {uj,., up, V1, ...,…

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Q: Let S be a finite set in a vector space V with the property that every x in V has a unique…

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Q: Prove that for any vector x, we have (a) x₂ ≤x≤√₂x₂. 1 (b) x ≤x≤x n

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Q: 24. Let S be a basis for an n-dimensional vector space V. Show that if v, v2. ...,v, form a linearly…

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Q: Let V be a vector space over F, and suppose that the list (v1, . . . , vn) of vectors spans V, where…

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Q: Let {u1, let B be a basis for V. Let S vectors of {u1,·· , Un} with respect to basis B. Prove that ,…

A: Given V is a n- dimensional vector space. Let B be a basis for V. Let B=v1,v2,…,vn. Let u1, u2, …,…

Q: Label the following statements as true or false, and give a brief justification If V is a vector…

A: True. If possible let S be linearly independent with n vectors but do not span V

Q: bination of (v₁,..., vn). Then we must have m≤n, and there is a replacement of m of the vectors u,…

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Q: If V is a 7-dimensional vector space and S is a set of 10 vectors, then the elements of S must be…

A:

Q: Let V be a vector space with inner product.  Suppose S is a subset of V and is linearly independent,…

A: We are given that V is a vector space with inner product and S is a subset of V. Also, S is linearly…

Q: Prove or give a counterexample: If v1, V2, .. Vm and w1, W2, .. Wm are linearly independent lists of…

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Q: Given the set S={v1,v2........vn} of n distinct linearly independent vectors in a vector space v.…

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Q: Let ẞ be a set of vectors in a vector space V with the property that every vector in V can be…

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Q: Prove that if S = {v1, v2, v3} is a linearly dependent set of vectors in a vector space V , and v4…

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Q: Every linearly dependent subset of a vector space V contains the zero vector. O True False

A: We have to see the given statements is true or false.

Q: Let {v1, . . . , vn} be a spanning set for the vector space V, and let v be any other vector in V.…

A:

Q: Let V be a vector space and u and w be vectors in V. Prove that {u, w} is linearly independent if…

A: Linearly independent sets: A set of vectors {v1,v2.......vn} is said to be linearly independent if…

Q: then each set If V (F) is a finite dimensional vector space of dimensionn (n + 1) or more vectors of…

A:

Q: Let V be a finite-dimensional vector space with a basis β, and let β1, . . . , βk be a partition of…

A:

Q: Let S ={v1 , v2 ,...,vn } be a set of nonzero vectors in a vector space V such that every vector in…

A: Given that, S=v1,v2,...,,vn⊂VAnd every vector in V can be written as unique linear combinations of…

Q: Let V be a vector space spanned by the vectors u, v,andw. Then which statement is TRUE? A. u, v,andW…

A: The statement 'A: U,V,W must be linearly indepent'  is a false statement since if V is of dimension…

Q: Let V denote a vector space over F. Suppose {B1, B2,..., B1, B41} is linearly independent in V and…

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Q: 17. Prove that if S = {v,,Vz, V3} is a linearly dependent set of vec- tors in a vector space V, and…

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Q: Let V be a vector space over R and let S = {v1,..., Um}. Suppose every element of %3D V can be…

A: Let V be a vector space over ℝ and let S={ v1 , v2 , ............ , vm } suppose S={ v1 , v2 ,…

Q: Let V be a vector space and let S = {v,…, Vn} be a set of vectors in V. Prove if the elements of S…

A: Linearly dependent set

Q: 5. Let V be a vector space over R, and suppose that v1, v2, v3, V4 are linearly independent vectors…

A:

Q: Let v1, v2 , v3 , and v4 be vectors in a vector space V. If {v1,v2 ,v3} is a linearly dependent set,…

A: First one proved using definition and second one proved using contradiction

Q: Let v1, and v2 be two vectors in a vector space V. Show that v1 and v2 are linearly dependent if and…

A:

Q: Let V be an F vector space of dimension n. Prove that, for k≤n the vectors V₁, V2, …, Vk are…

A: A set of vectors {v1,v2,...,vk} is linearly independent if the vector equation x1v1+x2v2+···+xkvk=0…

Q: Let β ={x1,...,xn} be a subset of Fn containing n distinct vectors, and let B denote the element of…

A: Assume that β ={x1,...,xn}  is a basis for Fn.Let B denote the element of Mnxn (F) whose jth column…

Q: Consider the vector space V with dimension = n, then any set of V with exactly n vectors should be a…

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Q: Let {v1, v2, . . . , vk} be a linearly independent set of vectors in a vector space V. Delete the…

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Q: Let V be a vector space that contains a linearly independent set {u1, U2, U3, U4}. Describe how to…

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Q: Let {u1, ·.., Un} be a list of vectors in an n-dimensional vector space V and let B be a basis for…

A: Given V is a n- dimensional vector space. Let B be a basis for V. Let B=v1, v2, …, vn. Let u1, u2,…

Q: Let V be a vector space, and let S be a subset of V. What does it mean when we say that S spans V? O…

A: We will use the definition of span of S to solve the given question .

Q: 4. In the vector space P3(R), let S = {p1(x), p2(x), p3(x)}, where P1 (x) = 63x° + 19x² – 37x – 41,…

A: (a) we see, p2(x) = -3p1(x), so S is not linearly independent. [Ans] (b) As S is not linearly…

Q: 5. Prove that for any vector x, we have (a) 1, <H<#H, (b)

A:

Q: 5. Let V be a vector space over R, and suppose that v1, V2, V3, V4 are linearly independent vectors…

A: Solution:Given:  V is a vector space over K and v1, v2, v3, v4 are linearly independent vectors in V…

Q: O Let u, v, w by any vectors of a vectors a vector space V and S={ v-ü, u-w, w-i }. Is S linearly…

A: The answer is given as follows:

Q: Let S= {a, a ,a_} be an orthonormal set of vectors in an inner product space V(F). If a vector B in…

A:

Q: 3. Show that if two vectors in the set {a1, a2,... , ak} are identical, then the set is linearly…

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Q: Let V (F) be à vector space and a,, az, ..., a, e V. If some az 2<k<n is a az-1 then prove that…

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Q: Let V be a vector space having dimension n, and let S be a subset of V that generates V. (a)  Prove…

A: (a) If we can find a subset containing n elements in S such that they are linearly independent, then…

Q: Let x, y, and z be vectors in a vector space V. Prove that if x + y = x + z then y = z.

A: if, x+y=x+z we use transportation and solve it

Q: Suppose that S = {v1, v2, v3} is a linearly indepen- dent set of vectors in a vector space V. Prove…

A:

Q: Let V be a finite-dimensional vector space, and let {v1, v2, ... vn} be any basis for V. (a) If a…

A: According to the given information, Let V be a finite dimensional vector space and,

Q: Let u and v be vectors in an inner product space V. Prove that ||u + v|| = ||u − v|| if and only if…

A: Given: Let u and v be vectors in an inner product space V. Prove that  ||u + v|| = ||u − v|| if and…

Q: Let V be a vector space, and let -u and -w be nonzero vectors in V. Prove that if ~u and -w are…

A: Let V be a vector space and let u and v be two nonzero vectors in V. To prove: if u and v are…

Q: In a vector space V, prove that Ov=0 for all ve V.

A: This quedtion from topic vector space of linear algebra.

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.