' Define an inner product on the vector space P2(t) as following 3 (p(), g(t) -Σ α.)g (a,) i=0 where ao, a1, a2, a3 = -3, –1, 1,3 Construct the orthogonal bases from p(t) = 1, q(t) = t,r(t) = t? space P2(t).

Q: Let C := {1, cos t, cos 2t, . , cos 6t} C V where V is the vector space of all real-valued functions…

A: Given: C:=1, cost, cos2t,…,cos6t⊂V. V=f | f:0,2π→ℝ is the Vector space. J=spanC To prove: C is a…

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B'A) Given the…

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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: 2. Let a, B, y be three vectors in the inner product space V½ (R) with the standard inner product…

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Q: V is a 3-dimensional vector space with basis By (V1, V2, V3) and W is a 3-dimensional vector space…

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Q: Compute and sketch the vector assigned to the points P = (0, 1, 1) and Q = (2, 1, 0) by the vector…

A: Solution:Given       P = (0, 1, 1) and Q = (2, 1, 0)F=<xy,z2,x>To sketch the graph of F

Q: 5. Show that if S = {v1,V2, .. .} is a basic for the space vector V and c is a non-zero ... ... ...…

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Q: 2.52 Prove that Ca, b] is a vector space.

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Q: Let V be the subspace of the vector space of all real-valued functions that is spanned by the setS…

A: If V is the subspace of the vector space of all real-valued functions that is spanned by the set S…

Q: 3. Let V be the set of column vectors of the form : x, Y, z E R, x > 0 with vector addition defined…

A: v={xy2;x,y,z<R,x>0

Q: If T is a normal operator on an inner product space V, then the characteristic vectors for T…

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Q: What is the divergence of the vector field f = x²i+ xyzj – zk at the point (1, 2, 3). Select one: а.…

A: f=x2i+xyz j-zK at (1,2,3)  div f=i ∂f∂x+j.∂f∂y+K.∂f∂z

Q: 6. [20p] Find a bases for the vector space V spanned by vector w = (1,1,0), w, = (0,1,1), w, =…

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Q: Consider the autonomous system of non-linear equations * = u(x, y), where u is a vector field with…

A: In this question, the classification of the equilibrium point is done.

Q: Find the divergence of the vector field. F(x, y, z)= 6x¹0i-10xy ¹0j O divF(x, y, z) = 60x⁹-100xy⁹ O…

A: To find the divergence of the given vector field.

Q: 3. The vector space V over R has a basis {u, v, w}. The linear map ø :V → R? has the property that…

A: First we find the image and kernel. Then verify the rank nullity theorem.

Q: 6) Consider the following vector spaces with corresponding basis: V₁ =R³ Bv₁ = {e₁,e2, е3} Bw₁ = {1,…

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Q: a) T: M, → R, where M, is the vector space of n by n matrices over a field, and T (A) = det(A).

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Q: b) Let Pn be a vector space consisting of polynomials of at most degree n. The standard basis for P,…

A: value of k is 0 see calculation below

Q: 4. Let B = {V1, V2, Vn} be a basis of a vector space, V. ... a) Prove just ONE of the following: •…

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Q: 3) Determine whether or not the following 3D vector fields are conservative: a) (y²-z³)i + (2xy +…

A: As per the company rules we are supposed to answer only the first question and rest can be reposted.

Q: ,Let Pn be the vector space of all polynomials of degree n or less in the variable r. Let D : P3 →…

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Q: 3. Consider R³ with the standard inner product (i.e. the dot product). Let {(} 1 1 U = span -1 1 2…

A: First we find the orthogonal basis of U, then find the third vectors which is orthogonal to the…

Q: Let C := {1, cos t, cos 2t, cos 6t} C V ...) where V is the vector space of all real-valued…

A: The given set is C=1,cost, cos2t, cos3t, …, cos6t⊂V. Where V is the vector space of all real valued…

Q: Theorem 7. Let T be a linear transformnation from space U (F) into a vector space V (F). and only…

A:

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B' A) Given the…

A:

Q: Consider the space of square integrable functions L² ([–1, 1]). Let B = {Pn|n = 0, 1, 2, ...} the…

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Q: 2. Let F = Q, K = Q(V5), and E = K(i, 5+ i). (a) Regard K as a Q-vector space (i.e. with Q as the…

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Q: If a vector field F = (x+2y + az)î + (bx – 3y – z) j+ (4x + cy + 2z) k is irrotational then the…

A: To find the value of a, b, and c if the below vector is irrotational.…

Q: 5. Let {u1, u2} be an orthonormal basis for an inner product space V. If r = c u1 +czuz is a vector…

A: According to our policy we’ll answer the first question since the exact one wasn’t specified. Please…

Q: Let B = {u, v, w, x, y} be a basis for vector space V, and let S = {u+ w, v, y} What is the…

A: Given basis is B = {u, v, w, x, y}

Q: Consider the vector space R with the Euclidean inner product (i.e. the dot product). Let W =u= y in…

A: Given Information

Q: 6. [20p] Find a bases for the vector space V spanned by vector w; = (1,1,0), w, =(0,1,1), w; =…

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Q: 2.4. Let V be a vector space and let v1, V2,..., Vn be a basis in V. For x = Ek=1 akVk, y = Prove…

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Q: Compute and sketch the vector assigned to the points P = (1, 2)and Q = (−1, −1) by the vector field…

A: A vector field for two dimensional space is defined as a function that gives a vector for each point…

Q: Theorem 7. Let T be a linear transformnation from a vector space U (F) into a vector space V (F).…

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Q: 10. Show that V = R² with the standard scalar multiplication, but addition defined by (7, , v.) +…

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Q: (a) In the geometry of this unusual inner product space (where ||x|| = /{x, x)), find the vector…

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Q: 6. Vector fields V and W are defined by V = (2x – 3y + z, –3x – y + 4z, 4y + z) W = (2x – 4y – 5z,…

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Q: Let B = {1, x, x², x³, xª}, which is a basis for the vector space of polyno- mials W = P4(R). Let V…

A: Let the best approximation of the function f(x)=sinx+cosx is g(x), where  g(x)=∑i=15f(x),eiei…

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

A: False

Q: H, ond H2 are subuedɔr spoces of vector space, Sp (H, U Ha) - H, +H2 show that I

A: Proof :-

Q: Q3. Let L: R2R2 be a mapping from the vector space R2 to itself defined by L(x , y)= ( 2x+y, x-3y)…

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Q: /- X be the vector Space of all Complex 2x2 and de fine T: x →x by Tx = bx, where Let matrices bEX…

A: Given :-  Let X be the vector space of all complex 2×2 matrices and define T : X →X  by Tx = bx ,…

Q: Theorem 7. Let T be a linear transformnation from space U (F) into a vector space V (F). Then T is…

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Q: Let T: V-->W be a linear transforma­tion between finite-dimensional vector spaces V and W Let B…

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: 7. Given below are three vector fields, F1, F2, and F3. Let C1 and C2 denote circles of radius 1 and…

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Q: Consider the vector space V = R 3 over the field of real numbers F = R. Do the following vectors…

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Q: 1) Verify that d/dx(fg) is a vector space over R, or explain why it is not.

A: The objective is to find whether the ddxf.g is a vector space over ℝ. Since it is known that…

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.