Show that if u and v are orthogonal, then || u * v || = || u || || v ||

Q: Find a and b such that v = au + bw, where u = ⟨1, 2⟩ and w = ⟨1, −1⟩. v = ⟨−7, −2⟩

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Q: Show that the list of vectors {r + 2, x² + 2, x³ – 3, x4 – x² + 1} is a linearly independent set of…

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Q: Prove that if ||u|| = ||v||, then u + v and u- v are orthogonal.

A: Given ||u||=||v||

Q: Show every vector a in R² can be written as a linear combination of and

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Q: If y is orthogonal to both and then y is orthogonal to every w in Span{u, v} V. O True False

A: We have to solve given problem:

Q: Let A = and v= Verify A(u + v) = Au +Av. 2.

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Q: Show that u and v are parallel if and only if u×v = 0.

A: The cross product of two vectors are defined as u×v=uvsinθ. Two vectors are parallel if the angle…

Q: Prove that u and v are parallel if and only if u × v = 0.

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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: For all u and v in R" with u + 0, show that u is orthogonal to v – u•v

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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: Find u × v and show that it is orthogonal to both u and v.u = (2, −3, 1), v = (1, −2, 1)

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Q: 3. If U and V are such that u; ='x; – A, v; =y, - B, then Cov (U, V) is :

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Q: Prove that if {Vi, V2, v3} is linearly independent, then so too is the set {u, u2, u3 where u = V -…

A: Introduction: Linear dependence and linear independence of a given number of vectors are very…

Q: Find u × v and show that it is orthogonal to both u and v.u = (1, −1, 1), v = (0, 1, 1)

A: Cross product is,

Q: Given the set of n non-zero vectors {v,v,,v} in Rm and n> m, prove that the set is a linearly…

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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: Let u = (−1, 1), v = (0, 1),and w = (3, 4). Find the vector x that satisfies u−2x = x−w + 3v.

A: u=-1, 1,  v=0, 1  &  w=3, 4

Q: - Prove that if {v1. V2, V3} is linearly independent, then so too is the set {u,, u2, us} where uj =…

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Q: Find u × v and show that it is orthogonal to both u and v.u = (−1, 1, 2), v = (0, 1, −1)

A: Given vectors

Q: Find a and b such that v = au + bw, where u = ⟨1, 2⟩ and w = ⟨1, −1⟩. v = ⟨−1, 8⟩

A: Given that v = au + bw. Substitute u =<1, 2> and w= <1, -1> in v.…

Q: Let u, , w e R. If i and o are orthogonal and and w are orthogonal, is it necessarily true that u…

A: We will check whether the given condition is true or not.

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

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Q: Find the projection ofv onto u

A: We have given two vectors v=-24 and u=-11 We have to find the projection of the vector v onto u.

Q: Show that u and v are orthogonal if and only if ||u+v|| 2 = ||u|| 2 + ||v|| 2

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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} is linearly independent, then so too is {u+ v, u - v}.

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Q: H is the set of all vectors x = X2 in R³ with the property that X3 X1 + x2 = 0 and Xi – x2 + x3 = 0.…

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Q: Prove that u × (v + w) = (u × v) + (u × w).

A: Here the objective is to prove that

Q: The projection of v onto w is v=i+j+2k, and w=i+2j+2k 3 =(-i+j+2k) а. (i + 2j +2k) 5 -(i + 2j + 2k)…

A: Projection vector of a⇀ on the b→ is given by c→=a→·b→b→b^

Q: Let T={(a, 0): aER}, then T is Borel algebra.

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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  &  w=w1, w2, w3 be the vectors. Now, v+w=v1, v2, v3+w1, w2,…

Q: (à) If éách véčtór in V can be expressed uniquely as a linear combination of vectors in S, then S is…

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Q: Show that fi(r) = rª and f2(r) = r² +1 are orthogonal on the interval [–1, 1).

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Q: The projection of v onto w is v =i+j+2k, and w=i+2j+ 2k 3 (-i+j+2k) а. (i +2j+ 2k) (i + 2j+ 2k) 3 (i…

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Q: Find (a) u × v, (b) v × u, and (c) v × v. u = (−1, −1, 1), v = (−1, 1, −1)

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Q: Prove that u, v, w ∈ R3     and [ u x v, v x w, w x u] = [ u, v, w]2

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

A: Condition of dot product.

Q: Suppose u, v E L. Prove that ((u ^ (¬v)) V v = u v ((¬u) A v)). a.

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Q: Is it true that if x is orthogonal to v and w, then x is also orthogonal to v − w?

A: x is orthogonal to v and w. Then their dot product is zero. That is, ⇒x·v=0  &  x·w=0

Q: Let u and v be u = [3,0,3] and v = [1,1,4]. Verify that: |u x vl? = |u]?\v]? – (u · v)²

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Q: (a) Verify the Cayley-Hamilton theorem. (b)  Use the theorem to express A2 and A3 as linear…

A: Part (a), Let, A=4-112 Now, the characteristic equation of A,…

Q: Find u × v and show that it is orthogonal to both u and v.u = (12, −3, 1), v = (−2, 5, 1)

A: Given vectors u=12,-3,1v=-2,5,1 Their cross product is u×v=ijk12-31-251…

Q: Let f(x, y, z) = 2x(y+z), x uv, y = v w, and z = w. Find in terms of u, v, and uw.

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Q: (b) In each part, determine whether the given vector 2x - x2 +x + 3 € P3 (R) is in the span of S =…

A: i) 2x3-x2+x+3∈P3ℝ is in the span of S=x3+x2+x+1, x2+x+1, x+1 if…

Q: If u, v, and w are vectors in R° , where u is non zero and u X v = u X w, then v = w. Select one:…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.