Prove that if V is a two-dimensional real inner product space, then thecomposite of two reflections on V is a rotation of V.

Q: Show that a collection {(-∞,q) C R | q is rational } is a basis for a topology on R.

A: Since for your B part there is no information regarding the  topology homework 2 problem 4. Hence if…

Q: Prove that if T is a unitary operator on a finite-dimensional inner product space V, then T has a…

A: Since, T is a unitary operator, there exists an orthonormal basis consisting of eigen vectors of T…

Q: Is it possible to make Z into a vector space if we redefine addition?

A: To check: the possibility to make ℤ into a vector space.

Q: Prove that every vector space has a unique zero vector.

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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 suppose that x and y are orthogonal elements of V. Prove that…

A: Given:

Q: Suppose V₁, Vm, and W are vector spaces. Prove that L(V₁ × ... X Vm, W) are isomorphic to L(V₁, W) ×…

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Q: If X is an inner product space and A = {0} , then A- = {0}O AO

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Q: Let V be an inner product space. Show that if w is orthogonal to both uį and u2, it is orthogonal to…

A: Given V be an inner product space. w∈V is orthogonal to both u1, u2∈V. Therefore: w.u1=0 and w.u2=0.

Q: Show that the vector spaces V = M2 and W = R4 are isomorphic

A: when dimensions of v is equal to dimensions of w then v and w are isomorphic solution is given in…

Q: In an inner product space, if , is not orthogonal to , and v is not orthogonal to w, then ū is not…

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Q: Show that the element 0 in a vector space is unique

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Q: Let V be the set of ordered pairs (a, b) of real mumbers with addition in V and scalar multiplicat…

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Q: Suppose v1, v2, v3, v4 span a vector space ?. Prove that the list v1 − v2, v2 − v3, v3 − v4, v4 also…

A: Let V be the vector space Since v1,v2,v3,v4 span the vector space V.

Q: Let T and U be self-adjoint operators on an inner product space. Prove that TU is self-adjoint if…

A: It is given that T and U are two self-adjoint operators on an inner product space.

Q: Let β be a basis for a finite-dimensional inner product space. (a) Prove that if <x, z>= 0 for…

A: Let β be a basis for a finite-dimensional inner product space.

Q: Every finite dimensional vector space is

A: We have to prove the given statement.

Q: 19. Prove that the set {(1, 0, 0), (1, 1, 0), (1, 1, 1), (0, 1, 0)} spans the vector R' (R) but is…

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Q: Show that the set is a sub-vector space of space R3 and find the generator set of this sub-vector…

A: A typical vector in H = (2t, t - k, -2t + 3k) = (2t, t, -2t) + (0, -k, 3k) = t(2, 1, -2) + k(0, -1,…

Q: Consider the set X = {a, b, c, d} with the discrete topology, construct a basis of space that has no…

A: Consider the set X=a,b,c,d with discrete topology. Consider the basis of this set as follows.…

Q: Let AABC and AA'B'C' be two triangles in the affine plane. Prove there is a unique affine map a such…

A: As we just have to prove the existence of an affine transform α , we need have to find the exact…

Q: Every finite dimensional vector space is algebraically reflexive. Prove that.

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Q: snip

A: Theorem:                  Suppose <v,w> is an inner product on a vector space V.  Then…

Q: Theorem: Let A and B be R-dibmodules af left R-modules M and Nrespectively. Then, MXN AXB

A:

Q: Let V and W be finite-dimensional inner product spaces. Let T: →W and U: W →V be linear…

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Q: Prove that if V is a two- or three-dimensional real inner product space, then the composite of two…

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Q: Show that the set of real polynomials of n – degree (Pn) is a vector space with traditional addition…

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Q: Let V be a K-vector space and U1, U2, U3 three sub-vector spaces of V. Show that: dim(U1) + dim(U2)…

A: first of all we must know about the dimension of the vector space.what is the dimension of the…

Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

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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: Find the orthogonal decomposition of v with respect to W

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Q: 9) Suppose u, v € V, where V is an inner product space, and ||u|| = ||v|| = 1 and (u, v) = 1. Find u…

A: I have used the property of norm,  IIxII=0 off x=0

Q: Let V be an inner product space. Show that if w is orthogonal to each of the vectors u1, U2, ·..,…

A: Inner product space: Let V be a vector space , then  v is said to be inner product space if  it…

Q: Let T be a normal operator on a finite-dimensional inner product space. Prove that if T is a…

A: By definition, given that T is a projection.

Q: The vector spaces M2, 2 and M5,1 are isomorphic to each other ylgn ihi

A: False

Q: 4) Suppose that T is an operator on an inner-product space V such that T2 = T. Prove that T is an…

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Q: (ii) Prove that W is a vector space of R°.

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Q: Let V = Rº. Give a definition of addition + or scalar multiplication on V that makes (V,+,-) unable…

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Q: (1) (a) Suppose V is a real inner product space and u, v e V. Prove that if ||u + o||2 = ||u||? +…

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Q: Let V be a finite dimensional inner product space and T a non negative operator on V. Then T has a…

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Q: Prove that (Pn, +, ·) is a vector space.

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Q: If V1, V2, V3, V4, and W are all vector spaces.  Are L(V1x...xV4, W) and L(V1, W) x...x L(V4, W)…

A: Given that: V1,V2,V3,V4,W=Vector space The objective is to verify the given statement. Using the…

Q: Remark: If (V, = « +p vi " = a + p

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Q: 1 Find the orthogonal projection of b=|2| onto W=Span| 1 using any appropriate method.

A: First we calculate a orthonormal basis in W.

Q: Let V be a real inner product space of dimension 2. For any x, y ∈V such that x ≠y and ||x||= ||y||=…

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Q: Verify that P,(R) = Eax | a, eR}, with addition and scalar multiplication !! 1-0 defined by Ź(a,…

A: let  k1  and k2  be two scalar  and  ∑aii=02xi   ∈P2ℝif P2ℝ is vector  space with given operation…

Q: Prove that the relation of orthogonality in.an inner product space is symmetric.

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Q: Let V be an inner product space with dimV = 2, and suppose that L= {u+v, u+2v} is a linearly…

A: Let V be an inner product space with dimV=2 and L=u+v,u+2v is a linearly independent subset of V.…

Q: If W is the sub space can find and orthogonal basis for a W

A: Given set is  w=span1-1100,-10001,10011,10001 Let  v1=1-1100, v2=-10001, v3=10011, v4=10001 Let us…