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 S be the subset of R2 consisting of all vectors of the form (a, b + 3). a) Show that (5, 2),…

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Q: Let u and v be non-zero vectors in 3-space. Prove that u and v are perpendicular if and only if |u…

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Q: Let 71, 72, 73 be any vectors in a 3-dimensional space. Determine whether the following statements…

A: You have put multiple questions without any numbering. So, according to the guidelines, we solve…

Q: determine whether each vector belongs to Null A where A=[1,-2,-1,0;0,1,3,-2;-2,3,-1,2]…

A: Since the result of AB  is not the zero vector, the vector B does not belong to the null space of…

Q: In a vector space V, prove that (-1)v= -v for all v E V.

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Q: Show that W is a vector space.

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Q: Let S = {u1, u2,..., uk } be a set of vectors in a vec- tor space and let T = {v1, v2,..., vm},…

A: Let V be the vector space let S=u1,u2,......uk be a set of vectors in V let T=v1,v2......vm be the…

Q: If A is m X n, prove that every vector in null(A) is orthogonal to every vector in row(A).

A: Let A be an m×n matrix. We want to prove that every vector in the null space of A is orthogonal to…

Q: Show that the set is linearty dependent by finding a nontrivial inear combination of vectors in the…

A: we have to show that the set is linearly dependent by finding a nontrivial linear combination of…

Q: Let V be a vector space, and let 01, 02, 03, 04 e V. Prove that Span (1,2,3,4} = Span (1-2, 2-U3,…

A: The second set is { v1→-v2→, v2→-v3→, v3→-v4→, v4→ }. Each element v1,v2,v3,v4 in the first set is…

Q: C: Let A = (v₁,..., v4) be any list of 4 vectors in the vector space V. Let B = (v₁-V2, V₁-V3,…

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Q: Suppose v1, ..., vn are linearly independent and Span({v1, .., vn}) = V , then if W ⊂ V and W…

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Q: Suppose ₁, 2, 3 is an orthogonal set of vectors in R³. Let w be a vector in Span(V₁, V2, V3) such…

A: Given: v1, v2, v3 is an orthogonal set of vectors in ℝ5.           v1·v1=30, v2·v2=25.25, v3·v3=9…

Q: Suppose {v₁, ..., vr} is linearly independent but does not generate the vector space V. Show there…

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Q: Let x and y be two orthogonal vectors. Then prove that +

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Q: vector

A: (a+b)(v+w) = (a+b)v + (a+b)w

Q: Find a, such that the set w-(1+2r-x?,a+x+x,5-5x+7x) is linearly dependent in the vector space R Ob.…

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Q: 2. Determine if the following vectors constitute a linearly independent set.

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

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Q: Given any linearly independent set of vectors V = {v, V2, , }, prove that there is at least one…

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Q: Show that the set S = {(4,3, 2), (0, 3, 2), (0,0, 2)} is a basis for R* by showing that it spans R³…

A: Consider the provided question,

Q: Use Vector Set A (a1 and a2) with a vector from Vector B (b1, b2, or b3) to show that while this set…

A: Explained below

Q: Let S = V₁ = {V₁, V2, V3, V4, V5} where, ) 22 = , V3 = ₂V4 = V5 = 22 2 Find a basis for the space…

A: Given: S=v1, v2, v3, v4, v5 v1=13-12, v2=2702, v3=1212, v4=21-20, v5=1221 We have to find a basis…

Q: Let v and w be nonzero vectors in R³ such that the equation XXV = W is satisfied by at least seven…

A: Given v and w are nonzero vectors in ℝ3 such that x×v=w is satisfied by at least seven distinct…

Q: Suppose V={v1,v2,...,vn} is a set of nonzero vectors, and let                    x= a1v1 + a2v2…

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Q: Give a non-zero vector in the null space of A. 4 A = -4 -5 2 4-3 -55-3 -1 1-5 -3.

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Q: Let v1= 2, 5 , 3 ), v, = ( 1 , 1, 1 ) and va = (4, -2 , 0 ) are vectors in R3 then are .a Linear…

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Q: Suppose V₁, V₂, V3 is an orthogonal set of vectors in R5. Let w be a vector in Span(V₁, V2, V3 )…

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Q: S= {(1,1,1), (1,2,-3), (5,-4,-1)} is a set in R'. Show that a) Sis orthogonal. b) Write w = (1, 5,…

A: Orthogonal set is the set in which dot product of two vectors is zero.

Q: Suppose {V1, V2, V3} are linearly independent vectors in a vector space V. Find the number(s) k such…

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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.