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- Let L be the operator on P3 defined byL (p(x)) = xp'(x) + p''(x) Find the matrix B representing L with respect to [1, x, 1 + x2].Let L be the operator on P3 defined byL (p(x)) = xp'(x) + p''(x) Find the matrix A representing L with respect to [1, x, x2].Let T be the function T: R³ R2 given by - (1) - [* =[^="], T (a) Show that T is a linear transformation. (b) Determine the standard matrix for T. (c) Find a basis for ker(T).
- Consider a linear transformation T from R2 to R² for which (:) -}- T and T Find the matrix A of T. A =Determine the kernel and the range of the linear operator on R': L(x) = (x1 + x3, X2, 0)4) Determine whether the given set of functions is linearly independent on the interval (-x,). a) (z) = cos 2z, (2) = 1, 1,()= cos z b) (z)=z, 1,(2)=z-1, (z)=z+3 c) ()=e**, 1,(=)=e
- Orthogonalize {1, t, t²} in P₂ (R) with 2 1 (f,g) = [*, f(x)g(x) dx. -1 Then normalize each one so that its graph passes through (1,1). These are the first three Legendre polynomials. " " 2 }a) Prove the one point rule - 3-version: (3x)(x = t ^ A) = A[x := t] if x is not free in t. b) Prove the dual result 6 on p.176. Prove (3x) AvBC = (3x) ÂС V (3x) BC. (Hint: Translate first to Standard notation) c) Prove using the auxiliary variable metatheorem: + (3x) (A → B) → (Vx)A → (3x)BLet T: R² R² be given by Find the matrix M of the inverse linear transformation, T-¹. M = T(x) »=[7]× -3 4 -5 X.
- Consider the vector space V of all the real polynomials of degree less than or equal to three, p(x) = ax^3 + b x^2 + c x + d, and consider the linear transformation T: V ------> V given by the derivative, i.e., T( p ) = p' = 3ax^2 + 2 b x + c. Find the eigenvalues and eigenvectors of this transformation.1 - [5] and v₂ - [3]. = Let V₁ = [T(v₁)]B = ( be the matrix for T:R² R2 with respect to the basis B = {V₁, V₂}. Find [T(v1)] and [T(v₂)]B. [T(v₂)]B = ( i i and let → i 4 -1 ^-43 A = tel )4) Find the adjoint of the operator Lu = u" + 4u' – 3u, where u'(0) = -4u(0), u'(1) = -4u(1).