Let x = (2,1+i, i) and y = (2−i, 2,1+2i) be vectors in C3. Compute <x, y>,||x||, ||y||, and ||x + y||. Then verify both the Cauchy–Schwarz inequality and the triangle inequality.

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A: Statement (S1) proves that  I2 , A, A2  are linearly dependent.

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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

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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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Q: Let the set of non-zero vectors V1, V2,..., Vn in some vector space V be orthonormal: (Vi, V) = dij.…

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Q: (b) For any arbitrary vectors u, v and w, prove that (u x v) x w = u x (v x w) if and only if (u x…

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Q: Let r = (2,-i, 1), and y = (-i, 3,1 + i) be vectors in C3. Compute (x, y), ||x||, |ly|l, and…

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Q: Show that the vectors 2 -2 2 uj = 2 uz = 3 uz = 7 1 -5 -3 are linearly dependent. Find all r1, x2,…

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Q: _3. Let x be a convex combination of {y1, . . , yk }. Assume in turn that each y; is a convex…

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Q: 4. Prove that, for any vectors A, B and C, that a necessary and sufficient condition А x (Вx С) %3…

A: The solution is:

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Q: Let u = u1i + u2j + u3k, v = v1i + v2j + v3k, and w = w1i + w2j + w3k be vectors in R3. Show that

A: given: u=u1i+u2j+u3kv=v1i+v2j+v3kw=w1i+w2j+w3k we have to show that: u×v.w=u1u2u3v1v2v3w1w2w3

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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

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Q: -2 Show whether or not the vector bs is a linear combination of the vectors a, = az = and az = that…

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Q: If v is a vector in R", and if k is any scalar, then: (a) ||v|| 20 (h) x = 0 if and only if y = 0

A: Given: v is a vector in ℝn and k is any scalar To show that  a) v≥0 b) v=0 if and only if v=0

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Q: Prove that for any vectors u, v, and w in R2 or R3, we have u ·(v + w) = u · v + u · w.

A: Let u=u1, u2, u3  ,  v=v1, v2, v3  &amp;  w=w1, w2, w3 be the vectors. Now, v+w=v1, v2, v3+w1, w2,…

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

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Q: 5b- За Let W be the set of all vectors of the form Find vectors i and v in R such that W = span {i,…

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Q: Suppose V₁ is a vector of all entries 1 in R" space, which we call "1-vector". What is ||v₁||2 (12…

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Q: 3. For any vectors U, V and scalars a, b, prove that (a) (aU) x (bV) = ab(U x V)

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Q: (2) Given the vectors - -- -- --- 1 V1 = V2 = V3 = and w = are there scalars a, b, and c such that…

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Q: 1 Compute the norm of the vector v = 1 ||| =

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Q: ind the norm of x and the unit vector u in the direction of a if -5 * = ||| = %3D 2 2

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Q: If x = (x1, x2)T , y = (y1, y2)T, and z = (z1, z2)T arearbitrary vectors in R2, prove that xTy = yTx

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