We showed in class that B = {2-3x+x², 6-7x+x³} is a basis for the subspace U={p E R3[r]|p(1) = 0, p(2)=0}. (a) Extend B to a basis of R3[r]. (b) Find a subspace W such that R3[r] = UW. (c) Find the unique decomposition of h(x) = x³ - 2x² + 3x -3 determined by the direct sum [that is, write h(x) = u(x) +w(r), with u EU and w EW]

Q: Which of the following sets of polynomials form a basis for P₂? (i) P₁ = 1 + 4x + 6x², P2 3+x = (ii)…

A:

Q: Let B = {B]. A} mation x→→ Ax. =[%] be a basis for R2 and A = Find the B-matrix for the transfor-

A: Given that The transformation is

Q: {0.0 (a) Find an orthogonal basis of W. Show your work. (b) Let v = [1 2 3 4]. Compute projw (v).…

A:

Q: Use the Gram-Schmidt process to find a orthogonal basis for Row (A) where 3 0 3 00 0-2 -2 0 0 -4 -4…

A:

Q: (4) V2 = A basis for R³ consists of the three vector V₁ = , V3 = 1 2 3 " (You do not have to verify…

A:

Q: 5. (a) Consider T = LA: R³ - where A is the matrix A = R¹, T(x) = Ax 1 0 0 0 1 0 000 000 (In the…

A:

Q: 3. Let A be a n x n matrix with distinct and positive eigenvalues. For each i, 1 ≤ i ≤n, let v, be…

A:

Q: Consider P2 (R) together with inner product: Find an orthogonal basis for P2 (R). (p(z),9(2) = Г'р…

A:

Q: = Complete the following truth table. Use T for true and F for false. You may add more columns, but…

A:

Q: Consider the linear transformation T:R² → R³ given by the following. (Enter your answers as a…

A:

Q: Consider the bases for M2x2(R) given by a' : and a the stan- dard basis. For the transformation T :…

A:

Q: given the linear transformation T: R³ → R² where T (ED) - (₂) T (D-3 (²) and 7(i) = (²). T Find the…

A:

Q: Thank

A: Solution.

Q: Let L: R2 R2 be defined by L 4*]-[***] represents the coordinates of v in the standard ordered…

A:

Q: The matrices A₁ = [18], 42 = [8], [8]₁4 = [] A3 = = form a basis for the linear space V = IR²X2.…

A:

Q: 5J Let the sub space VCR³ is given by X1 X2 X₁ + 3×₂ +2×3=0} V= Find a x3 basis of V

A:

Q: t: R³ R³ (x, y, z) → (z − x,x+y, z−y). a) Find the matrix of t with respect to the standard basis…

A: One has to find the Imafe and Kernel of the transformation.

Q: Let P2 denote the vector space of all one-variable polynomials of degree at most 2. Let B be the…

A:

Q: For this actvity, find the matrix representation [7] for the linear transformation 7 : R³ → R²…

A:

Q: The matrices 4₁ = [18] · A₂ = [od] · , A3 = [0].A 0 -[9] 0 A4 = → R2 form a basis for the linear…

A:

Q: B = (v1, V2, V3) is a basis of the vector space V and T: V → V is a linear transformation which…

A:

Q: (b) The row vectors of an n x n upper triangular matrix form a basis in R" if all entries in the…

A: The objective of the question is to determine whether the row vectors of an n x n upper triangular…

Q: (a) Show that {v1, v2, V3} are linearly independent, where 1 V1 = 1 and v2 = and v3 = -2 Hence {v1,…

A:

Q: 1) We write vectors horizontally for this problem. Let U be the subspace of IRS defined by U =…

A: