Question (7) Suppose {X} is a sequence of points (vectors) in R" converging to a. (a) Prove that ||XK|| → ||a||. (b) Prove that if b E R" is any vector, then b. Xkb.a.

Q: Two vectors u and v in space are if and only if u X v= 0

A: Given vectors in space are u and v The vectors cross product is u x v = |u| * |v|* sin(angle between…

Q: Suppose that i and v are vectors in R". If ū · ū = 4 and ||v|| = 9, then || 18

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Q: 5. Find a vector æ € R³ such that ||Ax – || is minimal, where [1 1 -2] 1 and b:= 2 1 1

A: By least square approximation ,      To minimize Ax-b the vector x is given by,…

Q: (c) Show that if a span{b₁,b2, ..., b;} then bib{T bi bi (0.0.3) Hint: You can prove the…

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Q: Two vectors M and N are defined as M<-c(3, 2, 4) N <- C(1, 2) What will be the output of vector Z…

A: It is given that the two vectors M and N :  M &lt;- c(3,2,4) N &lt;- c(1,2) Z &lt;- M * N Objective…

Q: Let u and v be vectors in R". Show that ||u – v||? – 2||u – 2v||2 + ||u – 3v||2 = 2||v||? %3D

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Q: 1 1 1 Let A 1 1 1 1 1 1 3/5 -7/5 is in null(A), or 4/5 Either show that the vector show that it is…

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Q: Q2 (a) Let n E N and let ở, w e R". Show that if ở · w = 0, then || ở + w||2 = || ||² + ||w||². (b)…

A: This question can be solved using the concept of dot product of two vectors.

Q: The following limit lim Select one: O None of them 8x³y³ (z,y) →(1,2) 2x - y O equals 12. O equals…

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Q: -2st] Let W be the set of all vectors of the form 3s + 5t t - 2s H v = Find vectors ū and 7 in R³…

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Q: 3.) Prove that the left singular vectors of A one the night singular vectors of A".

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Q: 2. Let a, b and c be vectors in R". Prove that (a) a and (a c)b - (a - b)c are perpendicular; a b…

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Q: Let a and b be vectors in R" such that ||a||=|| b||=2 and (a,b)= — . Then ||a - b||= ...

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Q: Find the terminal point Q of the vector v = 7i-j+3k given the initial point is P(-2, 3, 5) О (5, 2,…

A: In this question we have to find the terminal point of Q.

Q: [ - 6 15 Let A Find a vector in R? that is not in the span of the columns of A. 10 25

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Q: Suppose u, v and w are non-zero vectors in R" such that proj(-2v) a and proj(5w)= b. Use the…

A: Projection vector of a→ in the direction of b→ Proj b→a→= a→.b→b→×b^=a→.b→b→b→2

Q: be rational numbers. Show that R is a vector space over the rational numbers Q, using standard…

A: We proved all the properties of a vector space.

Q: 3. [6 points] Consider the vector space V = P, over the field of real numbers F = R. Is the vector 4…

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Q: 2. Let a, b, c, and d be algebraic vectors in R3, For parts (a) and (b) below, state which theorems…

A: The dot product of the two vectors A→ and B→ is given by the following: A→·B→=||A→||·||B→||·cos θ…

Q: R = (-1,5,4,6,0); K(2,-2,1,0,8) 1) d(R, K ) ; 2) ||R|| ; 3) ||K||

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Q: (b) (5 marks) Let u = 3i - 5j, v = -2i + 7j, Show that (u +v) (u - v) =||u||^2- ||v||^2

A: It is given that u=3i-5j and v=-2i+7j It is known that if A=ai+bj then A=a2+b2 Now find out the…

Q: Let A = [[3,1,1,2,2],[-3,-2,4,2,2],[-5,5,4,-1,-2]]  Give a nonzero vector x in the nullspace of A.

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Q: Find the norm of i and the unit vector i in the direction of a if ||7|| =

A: Solutions

Q: Let H be the set of all vectors of the form 48 -S -S -3s Find a vector 7 in R³ such that H =…

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Q: Find the norm of a and the unit vector ū in the direction of a if -5 -3 1 -2 |||| i = ||

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Q: Determine whether the following sets of vectors in R'(R) are L.D. or L.I. 1) {(1, – 2, 1), (2, 1, –…

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Q: For any vectors i, vin R', we have |a+ = |u + |). True O False Reset Selection

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Q: Let 7 and w be vectors in R". Prove that If |7| = |7| = | 7 + |, then |7 - wf=3|7

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Q: Let u, v and w be three vectors in R³. Which of the following is not defined for all u, v and w?…

A: Here u , v and w are three vectors in R3.  We find which of the given is not defined for all u , v…

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

A: Given: v1 is vector of all entries 1 in ℝn space. To find: v12 for n=16

Q: 7 Let a and b be vectors in R" such that ||a||=||b||=2 and (a,b) = 1. Then ||ab||= ...

A: Let a and b be vectors in ℝn such that  a=b=2 and &lt;a,b&gt;=72 . Then we have to find the value of…

Q: Let A and B be vectors in R" and let c be a real number. Then (1) ||A|| 2 0, and || A|| = 0 if and…

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Q: Suppose that u is a unit vector which is orthogonal to some other vector . Then u (u + v) + ||projzu…

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