Let V be a vector space over R with inner product <, >. Prove that V is a vector space over C with inner product defined as (u+iv,u'+iv') =(u,u')+(v,v')+i[{u',v)−(u,v')]

Q: Find one voctor in R' that spans the intersoction of U and W where U is the xy-plane-that is, U =…

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Q: Consider the inner product space (P2, (, )) with (p, q) = p(to)q(to) + p(t1)q(t1) + p(t2)q(t,),…

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Q: Find a subset ofbthebvectors v1=(0,2,2,4), v2=(1,0,−1,−3), v3=(2,3,1,1) and v4=(−2,1,3,2)that forms…

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Q: If (x1,y1) + (x2,y2) = (x1x2,y1y2) and  c(x1,y1)= (cx1,cy1) are vector spaces, verify if both…

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Q: Let (V, (, )) be an inner product space and let f, g be vectors in V. if || f|| = 4, ||| 5 and || f…

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Q: Show that, in an inner product space, there cannot be unit vectors u and v with &lt;u, v &gt; &lt;…

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Q: If V is vector space, with u = (u, , U2) and v = (v, , v2) with u +v = (u,+v, , u2+v2) and ku =…

A: u=(u1,u2) , ku=(3u1,0) k=2, u=(-1,2)

Q: 2) Write the vector v as a sum of proj span{.…...} (v) and proj spo span{m} orthogonal set V₁,..., V…

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Q: Show that v, = (1,0,–1) , v, = (1,0,1) , v3 = (1,1,1) span R’. %3D Express vector v= (4,2, 2) as…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: If V₁, V2 and v3 are 3 vectors in a vector space V such that -7v₁-9v2 = 0. V, and v₂ are linearly…

A: Given that v1, v2, and v3 are 3 vectors in a vector space Vsuch that -7v1-9v2=0. Then v1, v2,and v3…

Q: 5) Let V, W be finite dimensional vector spaces over F. (a) ( Prove that (b) Use this to prove that…

A: Question is wrong • please see the justification

Q: let W be an inner product space and let w1 and w2 be vectors in W. suppose ll w1 ll =5^1/2 and ll w2…

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Q: Let (V, (, )) be an inner product space and let v, w be vectors in V. If (v, w) = 7 and w = 5, then…

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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: Find all vectors u such that ux(1,1,1)=(8,-5,-3).

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Q: Suppose v1, V2, V3 is an orthogonal set of vectors in R. Let w be a vector in span(v1, v2, V3) such…

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

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Q: Find ProjS(u) where S is the space spanned by v1= (1/3)(2,1,2) and v2= (1/3)(-2,2,1). Find the…

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Q: What is the dimension of the vector space spanned by the vectors v1, v2? v1 = (6, -2, 4) and v2 =…

A: v1 = (6, -2, 4)  v2 = (14, 2,8)  There does not exist any real number c such that v1 = c v2 hence v1…

Q: Let V = R' and define vector addition as (x, y, z)+(x', y', z')= (x+x',y+y',z+z') (standard…

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Q: Verify all vector space axioms under the given operations (x,y,z)+(x',y'z') = (x+x'+1,y+y'+2,z+z'+3)…

A: The objective of the question is to verify if the given operations satisfy all the axioms of a…

Q: Prove that a(-3,2) and b (2,-3) form basis vectors in xy plane

A: a set is said to be basis if set is linearly independent and it spans the space.

Q: Show that VI = (1,0,1) , v, = (1,1,0) , v3 = (1,1,1) span R'. Express vector v= (2, 1, 3) as linear…

A: v1 ,v2 ,v3 ,...,vn span R^n if det(v1,v2,...,vn) is non zero

Q: Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1…

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Q: 9. Determine whether the vectors v₁ = (1,1,2), v₂ = (1,0,1) and v3 = (2,1,3) span the vector space…

A: We know that the dimension of vector space  is 3.If the given three vectors are linearly…

Q: Consider the vectors u₁ = [1,0,−1], _U2 = [1,0,0] and u3 = [0, 1, 1], v = [5, 1, 3] and w = =…

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Q: 11. Let (V,(, )) be a complex inner product space. For vectors v, w, set (v, w)r = 1/[(v, w)+(w,…

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Q: 49. Show that in the vector space V, (R), the vectors (1, 2, 3), (- 2, 1, 4), (- 1, – 1/2, 0) form a…

A: We prove that the set of vectors form a linearly dependent set.

Q: Suppose that u, u and w are vectors in a real inner product space such that (u, w) = 16, (v, w) = 5,…

A: Instead of u¯,v¯,w¯ , we are writing u,v,w. (for sake of convenience). Given: (i) &lt;u,w&gt;=16(ii)…