Determine whether the set of all ordered triples of real numbers with the op- erations defined below: (x. y, z) + (x',y'.2) = (x+x',y+y,z+z) k(x,y, z) = (x, 1, z) is vector space or not.

Q: Prove thatII = {(x, y, z)|x – 2y+ 3z = 0}is a subspace of JR³, %3D -

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Q: Let V be the vector space of functions from R into R. Show that f,g,h EV are independent where: f(x)…

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Q: Assume that Ax, y, z) is a vector field whose i component is only in terms of y and z, whose j…

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Q: The set ((x,y) : × 20, y 20) with the standard operations in R in not a vector space.

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Q: Show that if a vector field has the form F = ⟨ƒ(y, z), g(x, z), h(x, y)⟩, thendiv F = 0.

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Q: F  is the three-dimensional vector field defined by F(x,y,z) = ( z, x, y ).  In other terms,…

A: Consider the given information.

Q: Consider inner-product = | f(x) g(x) dx defined for vector space C[-1, 1], then for the function…

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Q: 27. Let S be the set of all vectors both of y in R' such that x + y -z = 0 and 2x – 3y + 2z = 0.…

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Q: Let F(x,y,z)=(−6xz2,6xyz,6xy3z)F(x,y,z)=(−6xz2,6xyz,6xy3z) be a vector field and…

A: Given, Fx, y, z=-6xz2, 6xyz, 6xy3zfx, y, z=x3y2z

Q: be a linear operator on a vector space V(F). If = 0, t ce oft and null space of t. Also given an…

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Q: Determine it this new def ot O wald make Va Vector Space

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Q: Let V be the set of all polynomials over Q of degree 2 together withthe zero polynomial. Is V a…

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Q: V=(x^2)yi-xyj+xyzk

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Q: Let V = {(x, y): x 20) with standard operation on R2. Is V a vector space?

A: Given set is  V={(x,y):x=>0 }

Q: Suppose n is a positive integer. Prove that 1 cos x cos 2x cos nx sin x sin 2x sin nx IT is an…

A: Given function is, f,g=∫-ππfxgxdx So there is multiple case arises: Case 1: fx=12π and…

Q: Let V and W be subspaces of R", and dim V and dim W are strictly less than n. Assume that no…

A: Given:- V and W are subspaces of ℝn              dim V<n, dim W<n              Also, there is…

Q: 3. Prove that Ov=0 for every vector v in a vector space.

A: We have to prove that 0v = 0 for every vector v in a vector space.

Q: Let fogec[agb] where c [a, b] Lt is the Vector sPace of the continuous functions overthe…

A: Let , f , g ∈ Ca , b  , here Ca , b is a vector space of the continuous functions over the closed…

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

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Q: Let V ((x,y): x 20) with standard operation on R2. Is V a vector space?

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Q: There is a vector field F(x, y, z) = (M (x, y, z), N (x, y, z), R (x, y, z)) , such that divF (x, y,…

A: We need to determine the validity of the statement: divF=div(curlF) for a vector field of the form:…

Q: Let F,G be linear maps of vector space V onto itself. Show that: (FoG) = G -1 F-1,

A: Given that F,G be linear maps of vector space V onto itself.

Q: Consider inner-product = ] f(x)g(x) dx defined for vector space C[-1, 1] , then for the -1 function…

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Q: Find the boundary of the sequence xn in the normalized vector space C [a, b], if it exists. 4 cos/3…

A: Solution: Here the sequence is xnt=4cosn3t5+cosnt and a=-5π,  b=-4π

Q: Determine the dimension of the vector space. R5

A: If V is a part of a finite set, then V is said to be finite-dimensional, and the dimension of V,…

Q: FBShow that these vectors (taken from a vector space consisting of all functions of the real…

A: The objective is to show that the vectors of the given set are not linearly independent.

Q: If V is infinite-dimensional and S-is an infinite-dimensional subspace, must the dimension of V/S be…

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Q: Find the curl of the vector field F(x,y,z) = x²zi – 2xzj + yzk and the point (2, -1, 3).

A: Let the given vector field is :

Q: O Show that whether the functions sint and cost in the vector space C[-n, 7]of continuous functions…

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Q: Let W is a finite dimensional subspace of an inner product space V and y is any vector in V. The…

A: Let W be a finite dimensional subspace of an inner product space V. Let T be an orthogonal…

Q: Let V = {(x, y): x 20} with standard operation on R2. Is V a vector space?

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Q: Let V = {(x,y) where y= 3x+1 x ER} with scalar multiplication defined by k(x,y)=(kx,y) is a vector…

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Q: . What is the basis and dimension of the vector space: SPAN {1, cos t, cos 2t}

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Q: Let V be n-dimensional vector space over c en for any n scalars ₁, ... , E F there-

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Q: Let X be the vector space of all ordered pa y = (n1, N2), · of real numbers. Show that norms on %3D…

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Q: Check whether or not the set V is vector space over R' with respect to the indicated operations. (x,…

A: The solution is given as follows

Q: What is the basis and dimension of the vector space: SPAN {1, cos t, cos 2t}

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Q: Let F(x,y,z)=(2xz^2,−2xyz,9xy^3z) be a vector field and f(x,y,z)=x^3y^2z (∇f=(  ,  , )). (∇×F=(  ,…

A: Given:F(x,y,z)=2xz2,-2xyz,9xy3zf(x,y,z)=x3y2z

Q: Let II be a plane passing through the origin, and let n be a non-zero vector orthogonal to II. Then…

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Q: Consider inner-product = | f(x) g(x) dx defined for vector space C[-1, 1] , then for the function…

A: The solution is given as

Q: let s={v1, v2... , V, } a set of vectorin vector space V such that C1V1 + C2 V2 + ... + C, V, = 0 is…

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Q: Suppose W1, ..., Wm are vector spaces. Prove that L(V, W1 × • × Wm) and L(V, W1) × ·…× L(V, Wm) are…

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Q: Let V and W be subspaces of R", and dim V and dim W are strictly less than n. Assume that no…

A: To prove: If V and W are subspaces of ℝn such that dim V < n, dim W < n, V is not contained in…

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: If W is a subspace of the vector space V = R, then the vector zero belongs to W: Select one: O True

A: A subset W of a vector space V is called a subspace of V if W is itself a vector space under the…

Q: There is a vector field F (x, y, z) such that curlF (x, y, z) = (divF (x, y, z) , (divF (x, y, z))²,…

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Q: Let R be the field of real numbers. Then {(x,y,z): x,y,z are rational numbers } is a subspace of…

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Q: : Verify the divergence theorem for A=4xi +(2Y+z Vector field ) | +(52-2x)k in the regim beunded by…

A: According to the given information, it is required to verify the divergence theorem for a vector…

Q: 2. Verify that the usual length of Vector in the has the plane properties in 3 dinensional Space…

A: Given: usual length of a vector in the plane or in 3 dimensional. To prove that this vector has the…

Q: Let H be a subspace of real space R. In terms of a geometrical definition of H, which of the…

A: Given:

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.