5. Let V be a vector space, and let x1,X2, ... , Xk E V. 5(b) Prove that if x1, X2, ..., Xk is linearly dependent, then either x1 = 0 or x;+1 € span{x1,..., X;} for some j : 1, 2, ... , k – 1.

Q: 30. In the vector space R2x2, c=[] A = 0. 0. B = 1 C -1 -3 are given. (a) Is {A, B, C} linearly…

A:

Q: Let A - ( ) a. Show that the vectors lz.A, A² are linearly dependent in Maz (R). b. Deduce that A"…

A: Statement (S1) proves that  I2 , A, A2  are linearly dependent.

Q: 9*) Let J and K be the Jacobian matrices of the functions y; = f:(x1, 82,. and their inverse…

A: Given: J & K are the Jacobian matrices of the functions yi=fi(x1,x2,....,xn) &…

Q: Let D={dj.,dn) and B-(b1.bn) for vector space U for x belong U and x = B[x]g=D[xD. P is transition…

A: We observe the role of the transition matrix here.

Q: Determine whether the given set S is a subspace of the vector space V DA. V R°, and S is the set of…

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Q: Let P2 be the vector space of all real polynomials of degree at most 2. Let p1, P2, P3 E P2 be given…

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Q: 4. Decide if the following statement is True or False: Let U, W, and X be subspaces of a finite…

A: Given - Let U, W and X be subspace of a finite dimensional vector space V such that U ∩ W = 0 and U…

Q: 15- Let A C B C V,and let V be a vector space, then if A is linearly dependent, then B is A:…

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Q: Let V and W be vector spaces, let T : V --> W be linear, and let { w1 , w2 , ... , wk} be a…

A: Given:          V and W are vector spaces and T : V → W be linear.          The set w1, w2, ..., wk…

Q: If T: V → W is a linear map between vector spaces V and W, T is one-to-one and onto, and a ba- sis…

A: Given that T:V→W is a linear map between the vector spaces V and W. Also, T is one-to-one and onto.…

Q: 8. Let x and y be vectors in 3-space, and suppose u is orthogonal to both x and y. Prove that u is…

A: Let x and y be vectors in 3-space and suppose u is orthogonal to both x and y  This implies  < x,…

Q: 6. Let U and W be 3-dimensional subspaces of a 4-dimensional vector space V, and suppose that U + W.…

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Q: Q4) Let C(R) be a normed space such that || f -sup{ Ifl,x E R}and T: C (R) -R define Tf)-If I. Is T…

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Q: (3) The subset of pairs {(r, y) E R²|ry = 0} in the vector space R².

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Q: 7. Let C[-7,7] be the vector space of continuous function over [-n, 1] with an inner product 1 (5,9)…

A: C-π,π is a vector space of continuous function, with inner product <f,g>=1π∫-ππfxgxdx show…

Q: 10. Let Y = {(x,y,z)| z=-x}. (a) Is Y a subspace of R³? Justify. (b) Is Y a vector space? Justify.

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Q: If V is a linear subspace of R", and x1 , ×2 , ... , Xn E V, then we must have span {x1, X2, Xn} =V.…

A: To determine that the statement, if V is a linear subspace of ℝn and x1, x2,....,xn∈V, then spanx1,…

Q: 4. Let B = {v1, V2, ..., Vn} be a basis of a vector space, V. %3D a) Prove just ONE of the…

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Q: Suppose V is a vector space and S, T = L(V, V) are such that Range Scker T=null T. Prove that (ST)²…

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Q: 3. (a) Let S = {v1, . . . , vn} be a linearly dependent subset of a vector space V , and let vn+1 be…

A: This question is related to linear algebra.

Q: a.) Let u = (2, 4) and v = (−3, 5). Compute u + v. b.) Let u = (2, 4) and k = 7. Compute ku. c.)…

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Q: Which of the following choices is not a subspace of vector space R? a. (x,y,z):z=x-y b.…

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Q: Theorem 7.35. Let X and Y be topological spaces. For every y E Y, the subspace X × {y} of X × Y is…

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Q: Prove that if S1 and S2 are orthogonal subspacesof Rn, then their intersection consists of only the…

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Q: 17. Prove that if S = {v,,Vz, V3} is a linearly dependent set of vec- tors in a vector space V, and…

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Q: Show that if S1 and S2 are subsets of a vector space V such that S1 ⊆ S2, then span(S1) ⊆ span(S2).…

A: Given:Show that if S1 and S2 are subsets of a vector space V such that S1 ⊆ S2, then span(S1) ⊆…

Q: Suppose u and v belong to a vector space V. Simplify cach of the following cxpressions: (a) E = 4(5u…

A: Given u,v belongs to vector space W

Q: Let S={2t, t- 7, t2 + 6} be a subset of the vector space P2. 1) S is linearly independent. (ii) Span…

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Q: 2. Prove that the set S is a subspace of the vector space V.(5 V = R3x3 and S is the set of upper…

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Q: II. Decide which of the following are subspaces of corresponding vector spaces: C R?

A: Vector space.

Q: Let V be the vector space of polynomials of degree <3 over R. Determine whether u, v,w E V are…

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Q: Let V and W be vector spaces and T:V→W be linear. Let {y1,…,yk} be a linearly independent subset of…

A: It is given that Let V and W be vector spaces and T: V→W be linear. Let {y1… yk} be a linearly…

Q: 4. In the vector space P3(R), let S = {p1(x), p2(x), p3(x)}, where P1 (x) = 63x° + 19x² – 37x – 41,…

A: (a) we see, p2(x) = -3p1(x), so S is not linearly independent. [Ans] (b) As S is not linearly…

Q: Prove that a subset W of a vector space V is a subspace of V if and only if 0 EW and ax+ y E W…

A: First proving the first part: If W is the subset of V and also the subspace of V Claim  : 0 W and…

Q: Let f1(x) = 3x and f2(x) = ∣x∣. Graph both functions on the interval −2 ≤ x ≤ 2. Show that these…

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Q: 1 If T is defined by T(x) = Ax, find a vector x whose image under Tis b, and determine whetherx is…

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Q: 5. Let V be a vector space, and let x1, X2,.. . , Xk E V. 5(a) Prove that if either x1 = 0 or x;+1 E…

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Q: Q3) Let L = Z* and |x ||= max {]x1 - x3l, x2l, 15x,1} Vx = (x1, X2, X3, X4) E Z*. Is (Z*, ) is a…

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Q: the following is true about the subset T={1+2x– x²,2+ 3x+x²,4+7x – x²} of the vector space R[x] of…

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Q: 2. Let V be an inner product space and v1, · · , Vn E V. For each of the following statements, •…

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Q: 1. Decide if the following subsets W C V are vector subspaces or not (justify) (a) W = x €RCR?. COs…

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Q: If V (F) is a finite dimensional vector space of dimension n then each set of (n+ 1) or more vectors…

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Q: Let V (F) be à vector space and a,, az, ..., a, e V. If some az 2<k<n is a az-1 then prove that…

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Q: Suppose that S = {v1, v2, v3} is a linearly indepen- dent set of vectors in a vector space V. Prove…

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Q: Therefore (0,0, 0) e W. Ex 6. If W, and W2 be any two sub-spaces of a vector space V, then inW, is…

A: We have to prove that intersection of two subspace of V  is also a subspace of V

Q: 3) Suppose U is a subspace of V with U + V. Suppose Se L(U,W) and S # 0 (so S is not the zero…

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Q: Let W1,W2,K1,K2, . . . ,Kp,M1,M2, . . . ,Mq be subspaces of a vector space V such that W1 = K1⊕K2⊕·…

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Q: If S and T are two subsets of a vector space V (F) and S cL (T) then show that L(S) CL (T).

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Q: 10. Suppose that V is a normed vector space over F. Let S = {x € V : ||x|| 1 is not an accumulation…

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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.