[o o 1 8. , Let A = |1 0 -1 0 1 Compute CA(A) e P3, the characteristic polynomial of 1 A, and [CA(A)]s, the coordinate vector of C4(A) relative to B where B = {1, A, X², X³} is the standard basis for P3.

Q: Find the matrix A of the linear transformation T(f(t)) = 6f'(t) + 8f(t) from P, to P2 with respect…

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Q: Let L: R → R³ be defined by L (a) Find the representation of L with respect to the nat- ural basis S…

A: Given: L:R3→R3 and is defined by, L100=110L010=201L001=101 It can be further defined as,…

Q: [1 0 1 A =| 1 1 0 2 1 15 - Find a basis for the zero space of the matrix 1 O A) [-1 1 1] O B) [-1 -1…

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Q: 59. In R4, given 2 nullspaces 1 2 00 1 V : |-1 0 -1 m 0- Find m such that dim(U n V) is max. Find…

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Q: Question 11 In the vector space P,, of polynomials of degree atmost two, the first column of the…

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Q: Let V = P2(R), give V the inner product defined by (f,g) = Lo s(1)g(t) dt, and let W = span {1 – x}.…

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Q: -2 Let y = what are the expansion coefficients of y in the basis {vo, V1, V2 , v3 } you found in the…

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Q: 1. For each of the following, use the Gram-Schmidt process to find an orthonormal basis for R(A): a.…

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Q: Let ẞ= {v1, ••• , vn} be an orthonormal basis for Rn. (a) Prove that, for any x and y in Rn, x · y =…

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Q: Find a basis {p(x), q(x)} for the vector space {f(x) E P2[x] | f'(-2) = f(1)} where P2[x] is the…

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Q: 2. Let B₁ be the basis 2²e², re², e² of V = span(2²e², zeª, z), and B₂ be the basis of that the map…

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Q: 3. Determine whether the set S = {1 – t, t, 1 + t2} a. Spans P2 b. Is linearly independent с. Is a…

A: Given set is  S = { 1-t ,t , 1+t^2}

Q: Let y = 2 1 what are the expansion coefficients of y in the basis {vo, V1, v2 , v3 } you found in…

A: In the Previous question, the vectors v(3) that is forms an orthonormal basis in R4, The vector are…

Q: (11) Find the dual basis {V4.V } of the stan dard basis of IR. Let T:IR → IR defined by T(xiy, z)…

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Q: Suppose A E R4×3 has the following QR factorization A QR V2 -V2 [1 0 1 0 0 0 0 -1 1 1 1 -1 R = 1…

A: We will first find the matrix A using the QR factorization given.…

Q: The coordinates of p(x)=5+3x+x² relative to the basis {1+x,2-x2,3+x+x²} is O (-1,1,2) none (0,1,1) O…

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Q: 9. Find a basis y1 (x), y2(x) of the space of solutions to y" = y such that Y1 (0) = 1, y (0) = 0,…

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Q: -2 Let T : R? → R² be defined by Tx = Ax where A = Find a basis 10. B for R² such that the B-matrix…

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Q: ii) Show that S = {(t – 1)³, (t – 1)², (t – 1), 1} is a basis of P3(t) where P3(t) stands for the…

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Q: .3 Let W = span{x, x' x³, x>} in C[0,1]. Use Gram Schmidt to find an orthogonal basis for W.

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Q: This is a textbook question, not a graded question   The first four Hermite polynomials of the…

A: We  have to find the basis for P3

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: Find a basis for the subspace a W = E R2x2 | 6a + 2b – c = d of R2x2- A basis for W is 1 0 0 Ex: 1 :

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Q: Let T : R? → R´be defined by [2 _5||X1 X2 1 6] [x2] а If B = is a basis for R“ such that |T'|B is a…

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Q: snip

A: (a).               Orthonormalize the set of the vectors                      v1 = 1111 , v2 =…

Q: Show that {e1,€2, - } is a Schauder basis for P, 1<p< ∞, where S1 ifi=n (E") en = %3D 0ifi#n.

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Q: Find a basis for Col B where [1 1 0 -2 0 0]

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Q: Extend {l + x, 1 + x + x2} to a basis for P 2.

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Q: Let n >= 3 be an integer. Let Vn(x) = {[x 1 1 1 ... 1 1]T,[1 x 1 1 ... 1 1]T, [1 1 x 1 ... 1 1]T,…

A: By the consequences of vector spaces we solve the given problem.

Q: Problem 1. A. Prove that B = {1+x, x+x?,1+x²) is a basis for P2 second degree polynomial space

A: Basis: Let V be a vector space, then a set of vector B={V1,V2,...,Vn} is called basis of set V is…

Q: What is the coordinate vector for the polynomial p(x) = 2 – x² + x'relative to the standard basis…

A: our objective is to find the coordinate vector for the polynomial p(x).

Q: 1) Let V = Q3 [x] and let W = Q2 [x]. Consider the transformation T: V → W defined by (a) d T(f) =…

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Q: 2. Let L: R3 → R³ and let 1 1 [L] = 3 -2 1 0 3 be the matrix of L with respect to the standard basis…

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Q: let B be the standard ordered basis for R2, and let 2 Use Theorem 2.23 and the fact that G )"-(; ) 1…

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Q: 9. Find a basis y1(x), Y2(x) of the space of solutions to y" = Y such that Y1 (0) = 1, y (0) = 0,…

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Q: Consider P2, endowed with the following inner product: 1 1 (p, q) : G P(0)q(0) + G P(1)q(1) + for…

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Q: Let L be the linear operator on R? defined by L(x1, x2) = (2x1 – 3x2, 21 + 4x2)*:\\ i). Find the…

A: We have to find three matrix A B and P as described in the question.

Q: Given that f is a polynomial of degree n in Pn, show that { f, f', f'', . . . , f(n)} is a basis for…

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Q: 6. Find a polynomial p(x) for which {r² – 1, x² +x +2, p(x)} is a basis of P2(R).

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Q: Consider the basis a = {1,1+x, 1+a+x ?} of P 2(R), and suppose 8 is another basis such that the…

A: Please look at step 2 for solution.

Q: Suppose A E R4×3 has the following QR factorization A = = QR V2 1 -1 1 |1 -1 V2 -V2 [1 1 0 1 -1 0 0…

A: Here, we need to find an orthonormal basis of A. First we will row reduce A to find the column basis…

Q: 2. Let B₁ be the basis r²e, ze, e of V = span(r²e², xe, x), and B₂ be the basis is H.6 of w that the…

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Q: 1.Assume (f,g) = | f(x)g(x)dx is the inner product for V = P, . Apply the Gram-Schmidt process to…

A: Orthonormal basis

Q: Let A = 2 1 4 so that A4(t) = (t–1)³. Define T : R³ → R³ by T() = Ax. %3D 1 0 3 1 by first (a) Find…

A: Given the matrix A=-10-4214103 so that △At=t-13 and define T:R3→R3 such that Tx=Ax (a) We have to…

Q: Q: Find a representation of Jp(t)> at a time t +0 using the ordered basis {|e1>, ]e2>, ]e3> }.

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Q: 1. Let {w₁, W2, W3} be an orthonormal basis for V. Given that v = between v and w₁ is 0 = 60° and…

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Q: Explain why S is not a basis for P2. S = {1, 8x, -6 + x², 3x} O sis linearly dependent. O s does not…

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Q: consider the following bilinear form qp defined on IR as follows : p(x, y) x1x2 +3x1Y2 + 3x2Y1+ Y1Y2…

A: note : As per our company guidelines we are supposed to answer ?️only first 3 sub-parts. Kindly…

Q: 12. Let B = {1+2x, x – x², x + x²}. (a) Show that B is a basis for P2. (b) Let p(x) =1+3x + x².…

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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.