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

Q: if x=u(1-v),y=(uv) then verify that a(u,v) a(x,y). =1 a(x,y)'a(u,v)

A: Given x=u1−v and y=uv We have to verify ∂u,v∂x,y⋅∂x,y∂u,v=1 Solve x=u1−v and y=uv for u and v, we…

Q: ủ (u x) = 0 for all u, v in R.

A: We will prove whether this is true or false as following.

Q: 3. For f(x, y, z) where x = u – v, y = v – w, and z = w – u: - Compute - Show that of - af af and af…

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Q: Find (u, v), ||u ||, || ||, and d(u, v) for the given inner product defined on R". u = (3, 0, 2), =…

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Q: prove thats. u.w V.W V.

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Q: Suppose v1, V2,..., Vm E V. Prove that {v1, v2, ..., Um} = (span(v1, v2, ..., Um))-.

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Q: Find (a) ⟨u, v⟩, (b) ||u||, (c) ||v||, and (d) d(u, v) for the given inner product defined on Rn.u =…

A: Given

Q: Prove that if w=f(x, y, z), x=r-s, y=s-t, and z=t-r then dw dw dw + + = 0. dr ds ôt

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Q: Let X, Y be two sets. Prove that X × Y C P(P(X UY )).

A: It is given that X,Y  be two sets. To show that X×Y⊆PPX∪Y Firstly, we define the set X×Y as, X×Y=x,y…

Q: Let u, EC prove that ū · v = v · ū น

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Q: et T E L(V,W), and U e L(W, X). Prove that UT : V → X is also linear

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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: Prove that V = Vw+ Vg; that is, E(X x - X )²= £ (x – X,)²+ £(x,. – X)? jk j,k j,k j,k

A: The solution

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

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Q: Find (u, v), ||u|l, ||v||, and d(u, v) for the given inner product defined on R". u = (1, 1, 1), v =…

A: Our guidelines we are supposed to answer only three subpart. Kindly repost other subpart as the next…

Q: Let F(x, y,z)=xi + yj+zk, f(x,y,z)=||F(x,y,z)||. Prove that: Vf" = nf"-²F F v(In f) =

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Q: Find (u, v), ||u||, V||, and d(u, v) for the given inner product defined on R". u = (-5, 0), v = (2,…

A: just I have used the definition of given inner product , using the definition for given vectors and…

Q: Find (u, v), ||u||, ||v||, and d(u, v) for the given inner product defined on R". u = (5, 0), v =…

A:

Q: if x=u(1-v),y=(uv) then verify that a(u,v) a(x,y)_1 a(x,y) a(u,v)

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Q: Find (u, v), ||u|l, ||v||, and d(u, v) for the given inner product defined on R". u = (3, 0, 1), v =…

A:

Q: Verify that the conclusion of Clairaut's theorem holds, that is, u, Uxy = "yx' u = x'y8 – y? With u…

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Q: Prove: if z = (V3)/2 + i/2, and %3D then z = i z" = 1 %3D

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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: if x=u(1-v),y=(uv) then verify that a(u,v) a(x,y). -=D1 a(x, y) a(u,v)

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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: Prove that f, uv v • dỉ = – §̟ vũu • dĩ

A: Whenever the force is conservative in nature the work done over the closed loop is always zero. i.e…

Q: Let T={[a, ) : a€R}, then T is not Borel algebra.

A:

Q: Find (u, v), ||ul|, |v||, and d(u, v) for the given inner product defined on RT. u = (1, 3, 0), v =…

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

A: Here the objective is to prove that

Q: . Prove that u, v, and ware all in span(u, u + v, u + v+w)

A: Given: Prove that u, v, and were all in span (u, u + v, u + v + w)

Q: For x, y, z, w > 0, prove that (x/z)+(y/z)+(z/w)+(w/x)≥4

A: It should by x/y, instead of x/z

Q: The denial of the proposition Vx E U, 3y E U, 3z E U such that x < y → x+ z == y is 3x E U, Vy EU,…

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Q: Find (a) ⟨u, v⟩, (b) ||u||, (c) ||v||, and (d) d(u, v) for the given inner product defined on Rn.u =…

A: u = (0, −6), v = (−1, 1), ⟨u, v⟩ = u1v1 + 2u2v2 We need to Find (a) ⟨u, v⟩, (b) ||u||, (c) ||v||,…

Q: Let X, Y ~ U[a, b] be independent. Define Z = max{X, Y}. Find the cdf of Z.

A: Let X, Y ~ U[a, b] be independent.

Q: Suppose u, VE Vand ||u|| < 1 and ||v|| < 1 . Prove it: V1- ||u||?/1- ||u||2 <1- (u, v)|

A: Suppose u, v∈V and u≤1 and v≤1 We have to prove that 1-u21-v2≤1-u,v Here, we have  u≤1 and v≤1 Also,…

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, and w are all in span(u, u + v, u + v + w).

A: To Determine: prove that u ,v and w are all in span (u,u+v,u+v+w). Given:  we have(u,u+v,u+v+w)…

Q: Prove that ||ū – w||< ||0|| + ||w|| where ū = (i, –2, 1+i) and ū = (1,2+i, –i).

A:

Q: Find (u, v), ||u|l, ||v||, and d(u, v) for the given inner product defined on R". u = (-4, 0), v =…

A:

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: Find (u, v), ||u|, |v |, and d(u, v) for the given inner product defined on R". u = (-5, 4), v = (0,…

A: We have given vectors u and v,  And we have to find a), b), c) , d)

Q: Construct a proof of the following fact: (Z V T) → P, Z, (P V R) → =(Q V T) t -(QV T).

A: Given:                                                           Z∨T↔P,Z,P∨R→¬Q∨T⊢¬Q∨T To do:…

Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

A:

Q: Let f(u, v) = u – v², u = xy, and v = z+ x. Find af in terms of x, y, and z. Provide your answer…

A: The given function is f(u,v) =u-v2   and u= xy,  v=z+x Now we use chain rule of derivative to find…

Q: Consider the set R. Prove: (i) VxER, x 0 = 0;

A: To prove: For any x∈ℝ, x·0=0.

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: Find (u, v), ||u||, ||||, and d(u, v) for the given inner product defined on R". u = (-8, 15), v =…

A:

Q: Prove that if w = f(x,y,z), x=r-s, y=s-t, and z =t-r then ôw ôw ôw + âr' as = 0. ôt

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Q: Suppose v1, V2, ..., Vm E V. Prove that {vı, V2, ..., Um}+ = (span(v1, v2,….., Vm))“.

A: Given,v1,v2....vm∈V To prove :      v1,v2....vm⊥=span(v1,v2....vm)⊥⇒v1,v2....vm  is a set of…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.