_3. Let x be a convex combination of {y1, . . , yk }. Assume in turn that each y; is a convex combination of { yi,1, the vectors { yi, j }. ... Yi,k; }. Prove that x is a convex combination of .. ,

Q: ii) Show that any four vectors oj, , 5, Vi in R are linearly independent.

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A: we have to find the vector in linear combination form

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Q: 5. Determine the number of vectors (x1,...,x„), such that each x; is either 0 or 1 and i=1

A: (x1,x2,...,xn) is a vector with xi is either 1 or 0.  Now, ∑i=1nxi ≥ k implies at least k of the…

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A: Let us assume that the vectors are linearly dependent.  So there must be x≠0 and y≠0 two real…

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Q: Consider the set of vectors S = {2 – x + 4x², 3 + 6x + 2x², 2 + 10x – 4x²} in P2. Show that the set…

A: use linear independent vector definition

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A: Given x and y are vectors in Rn we have to prove that x−y≤x−y Since, x=x Hence,  x=x−y+yAdd and…

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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…

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A: Given that: x=(1,1,1) and y=(0,3, -2) are two  vectors. Let us say,  w = (a, b, c ) is a vector…

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A: Answer :- A) Ans :-

Q: If x = (x1,x2)T, y = (y1,y2)T, and z = (z1,z2)T are arbitrary vectors in R2, prove that (a) xTx ≥ 0…

A: The objective is to prove the above expressions.

Q: 5) Suppose that the set of vectors {ỉ, 7, w} is linearly independent. Prove that Span({ū, T}) #…

A: Given that u→,v→,w→ is linearly independent. Prove Spanu→,v→≠Spanu→,v→,w→.

Q: ): Let v = 3 be two vectors in R. and v2 %3D Show that v = belongs to span{v1, v2}. 62

A: We know that if vector "v" is linear combination of "v1" and "v2" then vector "v"belongs to the span…

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Q: 1. Show that the following vectors are linearly dependent. Uz = U3= %3D -2 3.

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Q: 1. Consider the vectors a = (1, 1, 1) and y = (1, 1, 3) in Rº. Compute æ x y. 2. Show that, for æ, y…

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Q: 5. For which value of x do the following vectors form a linearly dependent set in R -1 y =|x, in- х,

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Q: Let u = (1, 3), v = (2, 1), w = (4,−1). Find the vector x that satisfies 2u −v + x = 7x + w.

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Q: A set of vectors {u1,... , Um} in R" is called orthonorma 1 if i = j 0 if i + j. 1 %3D U¡ · Uj

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

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Q: 6. Let y = write y as the sum of two orthogonal vectors, one u = -6 in Span {u} and one orthogonal…

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Q: Let u¯¯¯=⟨−3,−5⟩, v¯¯¯=⟨2,−2⟩, and w¯¯¯¯=⟨2,4⟩. Find the vector x¯¯¯ that satisfies 9u−v+x=2x+w

A: Given that: u=-3, -5v=2, -2w=2, 4

Q: 3. If x and y are two n-vectors, then prove that 1 (a) |x*y|< ]], \»l, - +=1 (Holder inequality) 1…

A: Given that x and y are two n-vectors. We have to prove the following :                          (a)…

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A: Answer

Q: 2. Let . t, t3, , be vectors such that t E span{t, , ta} then span(r, 2, a, a) = span{t, , } %3D

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Q: 2. Let i= and (a) Compute the length of the vector u.

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A: We will first write the formula for projection of a vector onto another vector b .

Q: Consider the vectors in M22 then S= {x1, x2, x3} is linearly dep Select one: O True O False

A: The solution is given below

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A: To show that if u is a vector in R2 or R3, then u + 0 = u.

Q: Let y = and u = 3 Write y as the sum of two orthogonal vectors, one in Span {u} and one orthogonal…

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Q: Let y→=[−8, −7] and u→=[−2, 6]. Write y→ as the sum of two orthogonal vectors, x→1 in Span{u} and…

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Q: 3. Show that that the vector u = |-5 is a linear combination of the vectors vị = |-3 3 2 V2 = -4 and…

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A: As we know that; The equation of the plane which is passing through a point Px0,y0,z0 and a normal…

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Q: Suppose wi and w2 are linearly independent. Let W = span(w1, w2) and u = projw v where v is a vector…

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