Let P, be the set of all polynomial functions in P(R) of the form ·+ anx", for some fixed positive integer n. 3. ao + a,x + Pn is a Show that ... subspace of P(R), and that {1, x, x², . . x"} forms a basis for P.

Q: Prove that every line through the origin in R3 is a subspace of R3.

A: To prove that every line through the origin in R3 is a subspace of R3.

Q: 1. If T is a normal operator on a finite dimensional inner product space V and if W is a subspace of…

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Q: 2. Let W be an (n – 1)-dimensional subspace of V. Show that V has a basis B satisfying Bn W = Ø.

A: Given W is an n-1 dimensional subspace of V which is n-dimensional. To show V has a basis B…

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A:

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A: Consider the example f(x) = {2 + x + x2}, this given polynomial has dimension 3 with the basis {1,…

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Q: Let L = {(x1, x2, x3, X4, X5) E R° | x1 X2 + 2x4 = 0, x2 – 3x4 = 0}. | Prove that L is a subspace of…

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Q: 2. Let W be an (n − 1)-dimensional subspace of V. Show that V has a basis B satisfying Bn W = Ø.

A: Please find the answer in next step

Q: 2. Let P3(R) be the vector space of all polynomials of degree 3 or less (and the zero polynomial).…

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Q: let V be the subspace spanned by {w1,w2,w3} .Find a basis {v1,v2} of V such that neighter v1 nor v2…

A: Given: V is spanned by {w1, w2, w3}  w1 = [-1 4 5]t, w2 = [0 -4 0]t, w3 = [2 -20 -10]t Consider the…

Q: 7 4. 3 and let W the subspace of IR spanned by u and v. Find a basis of W, the orthogonal complement…

A: The objective is to find orthogonal complement of W

Q: (a) Let V = F[x] be the vector space of polynomials over a field F. Show that the set S = {1, x, x²,…

A: As per our company guidelines we have solve only one question. Kindly repost other part in next…

Q: and let W the subspace of R" spanned by i and v. Find a basis of W-, the orthogonal complement of W…

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Q: Let S=v1,v2,v3,v4, where v1= (1,1,2,1), v2= (1,0,−3,1),v3= (0,1,1,2),v4= (0,0,1,1), and v5=…

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Q: Let S be the subspace of R4 spanned by x1 = (1, 0,−2, 1)T and x2 = (0, 1, 3,−2)T . Find a basis for…

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Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

Q: Let 1 -3 and let W the subspace of R' spanned by u and v. Find a basis of W-, the orthogonal…

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Q: 2. Consider the subspace im(A) of Rº, where 1 1 A = | 1 1 3 Find a basis of ker(A"), and draw a…

A: Given matrix A=111213 AT=111123

Q: Let f(x) = 6, g(x)= 8x + 7 and h(x) = -3x² + 2x + 4. Consider the inner product (p, q) = p(-1)q(-1)+…

A: A

Q: (a) Let V = F[x] be the vector space of polynomials over a field F. Show that the set S = {1,x,x²,…

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Q: Find an orthonormal basis for the subspace of C(R) generated by the functions {1, x, x²} with…

A: W e find an orthonormal basis for the subspace of C([0, 1]) generated by the function (1, x, x^2)

Q: 4. Let be B = {1,eª, xe²} a basis of a subspace E of the space of continuous functions, and let D.…

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Q: Prove that span (v1,v2,...,vn) is a valid subspace of R

A: We know that ,  spanv1 , v2 , .... , vn = a1v1 + a2v2 + ...... + anvn / a1 , a2 , ...... , an∈ℝ A…

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Q: Prove that Co is a Banach space by showing that it is a closed subspace of l

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Q: 2. Show that {1,x – 1, x (x + 1)} is a basis of the space of polynomials P2 (IR) and that W = {p(x)…

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Q: Find a basis for the subspace of R' that consists of all vectors perpendicular to both and 2 3

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Q: 3.) Determine the dimension of the subspace of P, consisting of all polynomials of the fom…

A: Dimension of subspace of p2 is 3 .

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Q: Let {u1 (x) = -3, u2 (x) = 12x, uz (x) = –8x² } be a basis for a subspace of P2. Use the…

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Q: (2) Find a basis for the subspace W%3D { (а+c, 2a — b, За — 2b — с, b + 2с) | а, b, с € R} of R4.

A:

Q: 3) Find a basis for the subspace of R that consists of the intersection of the planes X-3Y = 5Z and…

A: This is a problem of vector space.

Q: Let β be a basis for a subspace W of an inner product space V, and let z ∈V. Prove that z∈W⊥if and…

A: Let β be the basis for a subspace W of an inner product V, and let z € V.It is required to prove…

Q: Show that the set of all polynomials of degree at most 3 forms a subspace of Pn.

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Q: 5. Let S = {x + 1,x2 – 2, x – 1,3}. Find a basis for the subspace W that is spanned by S.

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Q: 7. Consider the subset S of R³, S = {(x, y, z) | 2x – y + 3z = 0}. Show that S is a subspace and…

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Q: Let {u1 (x) = 12, u2 (x) = 12x, u3 (x) = 8x² } be a basis for a subspace of P2. Use the Gram-Schmidt…

A: Orthogonal basis, therefore inner product will be zero for distinct two elements.

Q: 1- Show whether the set of polynomials a, +at +at+at ,where a, =a, +a, is a subspace of P3 or not ?

A: Let V be a vector space over the field F. A non-empty subset S of the vector space V is said to be a…

Q: 5. Let P2 be the vector space of polynomials of degree at most 2. Determine if the vector v = = 10x…

A: Consider the given vector v is in the given subspace. Consider the real numbers a, b such that the…

Q: .

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Q: 3. Given a nonstandard basis of a vector space of a second-order polynomial as R³ is given as B = {1…

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Q: Consider the vector space P3 of polynomials of degree at most 3 together with the zero polynomial..…

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Q: (a) Let V = F[x] be the vector space of polynomials over a field F. Show that the set S = {1, x, x²,…

A:

Q: Let be a basis for a subspace of R2, 12 the Gram-Schmidt process to find an orthogonal pasis under…

A: Gram-Schmidt orthogonalization process: Given a basis U1, U2, U3 to a vector space V,  Gram-Schmidt…

Q: 4. (a) Find a basis for the subspace H of R spanned by the following vectors: 00000 1 (b) What is…

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Q: Let {u1 (x) = -9, u2 (x) = 18x, uz (x) = –12x²} be a basis for a subspace of P2. Use the Gram- %3D 1…

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Q: Let f(x) = -1, g(x) = -7x + 2 and h(x) = 5x². %3D Consider the inner product (P, 9) = / P(=)q(=) dz…

A: Given fx=−1, gx=−7x+2 and hx=5x2 And p,q=∫04pxqxdx Gram Schmidt orthogonalization process is  ax=fx…

Q: ) Let W C R³ be a subspace with basis {{} 3 -5 -1 1 Use Gram-Schmidt process to produce an…

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Q: 5. Let W be the set of all vectors p(t) in P such that p(0) = 0. Show that W is a subspace of P and…

A: W be the set of all vectors p(t) in P2 such that p(0)=0.

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.