Let V be a finite-dimensional vector space with a basis β, and let β1, . . . , βk be a partition of β(i.e., β1, β2, . . . , βk are subsets of β such that β= β1∪β2∪·· · ∪βk and βi∩βj= ∅ if i≠j). Prove that V = span(β1) ⊕span(β2) ⊕·· ·⊕span(βk).

Q: Let V be a finite dimensional vector space with dimension n > 0. Let S = {V1, V2, Vn} be a family of…

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Q: Let S be a finite set in a vector space V with the property that every x in V has a unique…

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Q: 24. Let S be a basis for an n-dimensional vector space V. Show that if v, v2. ...,v, form a linearly…

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Q: Let P3 be the vector space of polynomials of degree at most 3, and let S = {p1(x), p2(x), p3(x), p…

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Q: Label the following statements as true or false, and give a brief justification If V is a vector…

A: True. If possible let S be linearly independent with n vectors but do not span V

Q: bination of (v₁,..., vn). Then we must have m≤n, and there is a replacement of m of the vectors u,…

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Q: Let W be a finite-dimensional subspace of an inner product space V , and let E be the orthogonal…

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Q: Show that if S is a basis for a vector space V then every vector in V can be written in one and only…

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Q: Let V be a vector space and S a subset of V with the property that whenever v1, v2, . . . , vn ∈ S…

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Q: Let V be a vector space with inner product.  Suppose S is a subset of V and is linearly independent,…

A: We are given that V is a vector space with inner product and S is a subset of V. Also, S is linearly…

Q: Let V be an n-dimensional vector space over F, and T:V→V be a linear transformation of rank 2. Prove…

A: T has rank 2 implies, for any two ordered bases the matrix representation of T has rank 2.

Q: (a) Let B = {V1, V2, . . . , Vn} be a basis for a finite-dimensional vector space V. Prove that for…

A: (a) If there exist such that, The take, Since B is a basis hence B is linearly independent. Hence…

Q: Let P3[t] denote the vector space of all polynomials in t of degree at most 3 with real…

A: To show that the given set is a basis for P3[t], it is sufficient to show that any element in P3[t]…

Q: Determine whether the set B is a basis for the vector space V. -1 V: B = 1 0 O B is a basis for V. O…

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Q: Define the following subset of R3 x1 E R³ : a'x= 0 = < x = x3 where a' = [-1,1,4]. Show that W is a…

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Q: Determine whether the set B is a basis for the vector space V. 0 1 B = 0 1 -1 1 V = M, 22' 1 0 1 1 1…

A: We will find out the required solution.

Q: Prove that if S = {v1, v2, . . . , vn} is a basis for a vector space V and c is a nonzero scalar,…

A: Given that is a basis for V and c is a nonzero scalar.

Q: Let M2x3(C) be the vector space of 2 x 3 matrices over C, with the usual rules of addition and…

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Q: Let V={a0+a1 x+a2 x2 | a0, a1, a2∈R} be a vector space of polynomial up to degree three, and…

A: We know that by definition of inner product x,y≥0 Take px=1+x+2x2, qx=1+x+3x2 then , p0=1, p1=1,…

Q: Let n be a positive integer, and let B = {v1, V2, ... , Vn} be a basis for a vector space V. Let c1,…

A: Solution

Q: Let B = {v1, v2, . ., vn} be a collection of n vectors in a vector space V over R. Suppose that…

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Q: 1. Determine whether the set B is a basis for the vector space V a. V = M22, B = [691-GJHG b. V =…

A: 1. Let V be a vector space over a field F. A set S of vectors in V is said to be a basis of V if S…

Q: Let B be a subset of a vector space V. Show that B is a basis for V ifand only if every member of V…

A: Let B be a basis of V such that B = {x1, x2, ..., xn} =&gt; Every element of V can be written as…

Q: any subspace U of a vector space E, if p: E- en pT is injective and Im (p¹) = U° = (Ker (p))º.…

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Q: Let V be a vector space and let S = {v,…, Vn} be a set of vectors in V. Prove if the elements of S…

A: Linearly dependent set

Q: Let V be a finite-dimensional vector space, and let B be a basis for V. Let S be a subset of V with…

A: Finite dimensional vector space and Basis of V

Q: Consider an n-dimensional vector space (V, R), that is, there are up to n eleme.ts of V that can…

A: We will first assume that vectors in W are linear dependent over R and prove that aour assumption…

Q: Let V be an F vector space of dimension n. Prove that, for k≤n the vectors V₁, V2, …, Vk are…

A: A set of vectors {v1,v2,...,vk} is linearly independent if the vector equation x1v1+x2v2+···+xkvk=0…

Q: 3(a) Prove that if the list X1, X2 is a basis for a vector space V, then the list ax1, bx1 + cx2 is…

A: Given x1, and x2 form basis of vector space V.  To show that ax1, bx1+cx2 also forms a basis of V…

Q: Let T be a function from P3 (F), the vector space of polynomials with degree at most 3, to P5 (F),…

A: Let, T be a function from ℙ3(F), the vector space of polynomials with degree at most 3, to ℙ5(F)…

Q: Let P3 be the vector space of polynomials of degree at most 3, and let S = {p1(x), p2(x), p3(x), p…

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Q: Let T be a linear operator on a finite-dimensional vector space V such that the characteristic…

A: Let T be a linear operator on a finite-dimensional vector space V such that the characteristic…

Q: If S is a finite set of vectors in a vector space V and a basis for V then it must be linearly…

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Q: Let V = {ao+a1 x+a2 x² | ao, a1, a2 € R} be a vector space of polynomial up to degree three, and…

A: The vector space of polynomials up to degree three is given by             V=a0+a1x+a2x2a0,a1,a2∈R…

Q: Let {V1, V2, ..., Vn} be an orthonormal basis for an inner product space V. Show that if w is a…

A: We will first write w as a linear combination of the vectors of the orthonormal basis. Then, we will…

Q: If W, and W2 are finite - dimensional subspaces of a vector space V, then W + W2 is finite –…

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Q: 5. Let V be a vector space over R, and suppose that v1, V2, V3, V4 are linearly independent vectors…

A: Solution:Given:  V is a vector space over K and v1, v2, v3, v4 are linearly independent vectors in V…

Q: Let V,W, and Z be vector spaces, and let T:V→W and U:W→Z be linear. a) If UT is one-to-one, prove…

A: It is given that V, W, and Z be the vector spaces, and let the two transformation will be T: V→W and…

Q: Let T be a linear operator on a finite dimensional vector space V, and Let W1,W2,.....Wk be T…

A: Assume that T is diagonalizable. Then there exists an ordered basis of V, v1,v2,...vk  such that…

Q: Find a basis {p(r), q(z)} for the vector space {f(z) € P3[r] |f'(3) = f(1)} where P3 [a] is the…

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Q: Let B = {v1, V2, V3, V4} be an ordered basis for a vector space V. Let Vị = span{V1, V2} and 1 0 1 1…

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Q: Let L1 and L2 be linear transformations from a vector space V into a vector space W. Let {v1,…

A: L1 and L2 be the liner transformation from vector space V into a vector space W. Definition.    Let…

Q: Let V be a vector space having dimension n, and let S be a subset of V that generates V. (a)  Prove…

A: (a) If we can find a subset containing n elements in S such that they are linearly independent, then…

Q: Let S be a linearly independent set of vector form a finite dimensional vector space V. Prove that…

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Q: Let W be the subspace of P2 defined by all p(x) in P2 such that p(x) = 0. (a) Construct a basis for…

A: (a) A polynomial p(x)=ax2+bx+c satisfies p(1)=0. Hence every p∈W is of the form,…

Q: Let V be a vector space of dimension n, and let B and C be bases for V. Then the product PB-c Pc-B…

A: Step:-1 According to a well known result,  Let V be a vector space of dimension n and C and D be two…

Q: Let V be a finite dimensional vector space of dimension m+n, and E be a projection on V, then show…

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Q: (a) Let B = {V₁, V2, ..., Vn} be a basis for a finite-dimensional vector space V. Prove that for…

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Q: Let W, and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and…

A: Given that  W1  and W2 are the sub spaces of a finite dimensional vector space V.

Q: Let W be a subspace of a vector space V. Under what conditions are there only a finite number of…

A: Let V be a vector space and W a subspace of V.If W has a finite number of elements,then it is…

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.