Using Different Methods In Exercises 19-22, find
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Multivariable Calculus
- Proof Let T be the function that maps R2 into R2 such that T(u)=projvu, where v=(1,1). a Find T(x,y). b Find T(5,0). c Prove that T is a linear transformation from R2 into R2.arrow_forwardProof Let A be a fixed mn matrix. Prove that the set W={xRn:Ax=0} is a subspace of Rn.arrow_forwardProof Use the concept of a fixed point of a linear transformation T:VV. A vector u is a fixed point when T(u)=u. (a) Prove that 0 is a fixed point of a liner transformation T:VV. (b) Prove that the set of fixed points of a linear transformation T:VV is a subspace of V. (c) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(x,2y). (d) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(y,x).arrow_forward
- Linear TransformationsIn Exercises 9-22, determine whether the function is a linear transformation. T:33, T(x,y,z)=(x+1,y+1,z+1)arrow_forwardShowing Linear Independence In Exercises 27-30, show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent. {eax,xeax}arrow_forwardCalculus Use the matrix from Exercise 45 to evaluate Dx[4x3xex]. 45. Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.arrow_forward
- Linear TransformationsIn Exercises 9-22, determine whether the function is a linear transformation. T:33, T(x,y,z)=(x+y,xy,z)arrow_forwardFinding the Inverse of a Linear Transformation In Exercise 49-52, determine whether the linear transformation is invertible. If it is, find its inverse. T:R3R2, T(x,y,z)=(x+y,yz)arrow_forwardSum of Two Linear Transformations In Exercises 67 and 68, consider the sum S+T of two linear transformations S:VW and T:VW, defined as (S+T)(v)=S(v)+T(v). Proof Prove that rank(S+T)rank(S)+rank(T).arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning