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 Find an orthonormal basis B for imgA.

Q: Find a basis of ker T. T: M22 → 1 M22; T( x ) = AX, where A = 0 0 O[1 0 0 1 0 1 1 0 1 0 1 0 0 -1 0…

A: The transformation T:M22→M22; T(x)=AX, where A=1100. We have to find the basis of kerT.

Q: 2- Let T:R> R? be the linear opentor def ined X- X27 X2 and let B={uisUz} be the basis for which and…

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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: Let P, have the inner product (P. 4) = | P(x)q(x)dx (p, q) Apply the Gram-Schmidt process to…

A: According to the given information it is required to calculate the orthonormal basis by applying the…

Q: The first four Laguerre polynomials are 1, 1 –t,2 – 4t + t², and 6 – 18t + 9t² – t3. Show that these…

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Q: A basis B of R² such that e1B and e1 + e2 B are orthogonal, where e1, e2 are the standard

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Q: Suppose T R-M22 is an isomorphism whose action on a basis for R* is as follows: -6 0 0 1 0. Find the…

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Q: Explain why S is not a basis for P2.S = {−3 + 6x, 3x2, 1 − 2x − x2}

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Q: 2. Prove that the following vectors |v1) = (1, 1, 0) |v2) = (1, 2, 0) |v3) = (0, 1, 2) span C3. Are…

A: Given that,                    |v1> = 1 , 1 , 0                    |v2> = 1 , 2 , 0…

Q: Find an algebraic description of the range of A by row reducing A^T and using the relation between…

A: We know that rank(A) = rank(AT). We need to find an algebraic description of the range of A by row…

Q: Given the basis {(1, 2,−2)T , (4, 3, 2)T , (1, 2, 1)T } for R3, use the Gram–Schmidt process to…

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Q: Let p1(t) = t and p2(t) = t + 1 be polynomials in P3. (a) Show that p1(t) and p2(t)} are linearly…

A: Given: p1t=tp2t=t+1 (a) We have to show that p1t and p2t is linearly independent. (b) We have to…

Q: Explain why S is not a basis for P2.S = {1 − x, 1 − x2, −1 − 2x + 3x2}

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Q: Consider the real quadratic form q : R’ → R, defined via: q(x1, x2, X3) = x + 25x; + 125x + 10x1x2 +…

A: Solution

Q: Say whether the polynomials P1 (x) = x³–7x²+4x, p2(x) = 3x³-x²+5x-1, p3(x) = x³+4x-x²+1 are linearly…

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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: Let A E Mpxn(F), and let B be the set of rows of A. Prove that B is a basis of F" if and only if…

A: The proof of given question is solved below.

Q: Find a basis {p(x), q(x)} for the vector space {f(x) e P2[x] | f' (7) = f(1)} where P2[x] is the…

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Q: The vectors 11 v1 = -4 V3 4 -5 form a basis for R° if and only if k +

A: The vectors are basis of R3 if they are not linearly dependent.

Q: snip

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

Q: 300311 By reducing to reduced echelon form, find a basis B of the null space, N(A) of A = 31151…

A: The given matrix is A=30031311510224003361. Find the basis B for the null space of A as follows.…

Q: The set B= is a basis of the space of upper-triangular 2 × 2 matrices. [5 1 with respect to this…

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Q: (b) Let W be the eigenspace of B corresponding to A = 1. Find a basis for W.

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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: 2. Let T : P2 –→ P2 be defined by T(a + bx + cx²) = a – 26 + (2b – 4c)x + 3 cx?. Find a basis C for…

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Q: 2. In each of the following, prove or disprove that B is a basis for P3, the vector space of…

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Q: (ii) Find the dual basis { V4. } f the stan dard basis of IR. Led T. in 3

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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: Explain why S is not a basis for P2.S = {−x, 4x2}

A: Given: S={−x, 4x2} To Explain: Why S is not a basis of P2.

Q: Let A ∈ Mn×n(R) be the matrix defined by Aij= 1 for all i and j. Find the characteristic polynomial…

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Q: Given any finite dimensional ve : E→ E, there is a basis (u₁,..., Un) with triangular matrix (in…

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Q: Let S = (2x, 1 - x, x2 +x 1,x+ 1}. Then PA P1 O sis a basis for the nbove Sis a basis for the above…

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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: Suppose that G E Fxn. Let X1, X2, ...Xe be a basis for Nul(G). Form a matrix H whose columns are X1,…

A: Note: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts…

Q: 4 El 4 Let Vi V2 3 and V3 = Use the Gram-Schmidt procedure to construct an ordered - 4 orthonormal…

A: Concept introduction: Gram-Schmidt orthogonalization process Let V be a vector space with an inner…

Q: 1 1 11

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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: 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: 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: Lemma 2.8 B is a basis of T iff

A: To prove that Β is a basis of Τ iff ∀Ο∈Τ ∃ Β0⊂Β: Ο=∪ ΒΟ

Q: Given that x, = – 1 and x1= 1. Then the basis of the first Lagrange polynomial Lo(x) and L,(x) are…

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Q: Consider B find [p(x)]B. {x2 + x, 2 + 2x, 1} an ordered basis of P2(R). If p(x) = 4.x² 2x – 6, %3D

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Q: Let S = {x; 2 – x; x² – 2x}. A. S is a basis of P2 B. S is a basis of Pa C. None of the above

A: The answer is:  (A) S is a basis of P2. Proved below.

Q: On P2(R), consider the inner product given by 1 (p.q) = | p(x)q(x) dx. Apply the Gram-Schmidt…

A: Given inner product is p,q=∫01pxqxdx Let u1=1, u2=x, u3=x2 Using Gram Schmidt Process to make…

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: 60. In R4, given a nullspace V = {(x1; X2; X3; X4) E R4|¤1 – *2+ x3 = 0 A x2 + x3 + x4 = 0}. %3D…

A: see below the answer and explanation

Q: Let 0 = V5 and K = Q(0). Two bases for K over Q are a = {1+0,1+ 0²,1+ 0³, 1 + 0ª} and B = 1+ 0²…

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Q: Let T: P2 that is, 7. > P2 be the linear operator T(p(x)) = p(2x+ 1), T(co+ C,x+ C,x²) = c, + c,(2x…

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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.