Let u, v, and w be vectors in R. Prove that (u+ w) v u v +w v.

Q: b) Prove that if b e R" is any vector, then b . X +b.a.

A: I have given the proof by using only definition of convergence of sequence

Q: Let A and B be vectors in R. Show that ||A|| = || B|| if and only if A+ B is orthogonal to Ã- B.

A: To solve the given orthogonality problem related to vector.

Q: Let v, x, y, z be vectors in R. Determine which of the following are well defined and which are not:…

A: We know that (i) vector product of two vectors is again a vector & (ii) scalar product of two…

Q: Let y= and u= 3 Write y as the sum of a vector in Span (u) and a vector orthogonal to u.

A: Describe y as the sum of vertices x1 in span u and x2 orthogonal to u         x1=α9, 3=9α, 3α and…

Q: а — Ь Let W be the set of all vectors of the form Find vectors i and v in R' such that W span {u,…

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Q: Prove that u is orthogonal to v - proju(v) for all vectors u and v in Rn, where u ≠ 0.

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Q: Prove that if x and y are vectors in R", then |||x|| – ||y||| < ||x- y||. X

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…

Q: Let u and v be two orthogonal and unitary vectors. Prove that ||u – v|| = v2.

A: Given:   Let u→ is a unit vector means that u2=1         v→ is a unit vector means that v2=1   u→…

Q: Show that there are no vectors u and v such that 11u 11= 1, ||v|| = 2, and uxv= 3.

A: If u and v are two vectors, and the angle between these two vectors is θ Then the magnitude of the…

Q: Let u, v, and w be linearly independent vectors. Show that (u+v), (u-v), and (u-2v+w) vectors are…

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Q: 1. Prove that if u and v are vectors in Rn, then |u·v| s ||u|||v|| where ju v| denotes the absolute…

A: As per guidelines we have to attempt only 1st question

Q: Let u, v, and w be three mutually perpendicular vectors in ℝ3.(a) Prove that u × (v × w) = 0.(b)…

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Q: 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 w)…

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Q: Suppose that u and v are vectors in 3-space such that u ||= 5 and u · v= 7. Find: (a) u · u = (b) v…

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Q: d) Prove that ||vxw||²+|v-w[? = ||v||²|w||² for all vectors v, w in R3

A: Prove the following

Q: Let u, v, and w be vectors in ℝ3. Prove:u × (v + w) = u × v + u × w and(u + v) × w = u × w + v × w

A: Like the scalar product, vector product of two vectors is also distributive with respect to vector…

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…

Q: Let z, w E C. Clearly prove that z + w = z+ū. Let u, v and w be vectors in C". Clearly prove that (u…

A: a) Let z, w ∈ℂ. Prove that z+w=z+w. Let z=z1+iz2 and w=w1+iw2. Add the two complex numbers:…

Q: Show that if u and v are n-vectors, then u · v = uT v.

A: Consider the n vectors  u=u1u2u3.un,v=v1v2v3.vn

Q: Prove that for any vectors u, v, w , the set span{u+v,u – v, 3u – 3v + 2w} contains u, v, and w .…

A: A vector v is said to be contained in span of set S  (span(S)) if vector v can be written as a…

Q: Show that the following vectors of V,(R) are LI : a, = (1, 1, 0); a, = (1, 1, 1); a, = (2,1, 3) %3D…

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Q: Let u, v, and x be the following vectors in R°. 2 1 u = V = -2 X = 4 Let W span{u, v}. Find the…

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Q: Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 6, (v,…

A: The norm of the vector, a is defined as:a=a, a. Using the linearity property of the inner product…

Q: Prove that u × v is orthogonal to both u and v.

A: Consider the two vectors u and v, Evaluate the cross product of the two vectors u and v,

Q: Prove that if vector u is orthogonal to both vectors v and w, then u is orthogonal to v + w.

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Q: Consider two vectors u and v in Rn such that u ∙ u = 39, u ∙ v = −3, and v ∙ v = 79. Evaluate (u +…

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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

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…

Q: Prove that the vectors v= and w= are orthogonal.

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Q: Suppose ~u and ~w are vectors in R^n . Prove that if ~u and ~w are parallel vectors in the same…

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Q: Show that the following vectors of V,(R) are LI : a, = (1, 1, 0); a, = (1, 1, 1); a, = (2, 1, 3) %3D…

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Q: Let u, v, and w be vectors in R^n (that is, u=(u1,u2,........,un) similar for v and w) Prove that…

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Q: Prove that if u, v, and w are vectors in Rn, then (u + v) ∙ w = u ∙ w + v ∙ w.

A: We have given three vectors  u, v, and w are vectors in Rn; we have to prove (u + v) ∙ w = u ∙ w + v…

Q: Let u = (-1,0,1), v= (3,2, 1) and w = (0,1, 2) . (a) Prove that the vectors u – v, u+ v and u x v…

A: Given : u=&lt;-1,0,1&gt; v=&lt;3,2,1&gt; w=&lt;0,1,2&gt;

Q: Show that if u and v are vectors in an inner productspace that satisfy the Pythagorean law||u + v||2…

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Q: Let u, v, and w be vectors in R". If u • v and u•w are both nonzero then v and w are not orthogonal.

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

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Q: Prove that for any vectors u, v E C", and scalar z € C, a) u· v = v•u b) zu v = z(u · v)

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Q: For any vectors U, V and scalars a, b, prove that (a) (b) (aU) × (bV) = ab(U × V) (aU) · (bV) = ab(U…

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Q: Find the norm of v, and a unit vector that is oppositely directed to v. (a) v = (3, 3)

A: The given vector is v = 3, 3

Q: Let u, v and w be vectors in Cn . Clearly prove that (u + v) · w = u · w + v · w.

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Q: Show that if u is a vector in R2 or R3, then u + 0 = u.

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

Q: For any vectors U, V and scalars a, b, prove that (aU) x (6V) = ab(U × V) (a) (b) (aU) - (bV) = ab(U…

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Q: Show that if u + vand ủ – v are orthogonal, then the vectors u and must have the same length

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Q: Prove that |A|B + |B|A is orthogonal to |A|B – |B|A, for any vectors A and B.

A: We need to show that |A|B + |B|A is orthogonal to |A|B – |B|A, for any vectors A and B. Here, |A|…

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: Show that if u and v are linearly independent vectors, then so are  u + v and u -v.

A: Given :           u and v are linearly independent vectors.   To prove :               u + v and…

Q: Let W = Span {V1,..., v,}: Show that if x is orthogonal to each Vj, for 1 < j < p, then x is…

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Q: Find two vectors v and w such that the projection of v onto w is v.

A: Given: There are two vectors v and w such that the projection of v onto w is v. We have to find that…