Kernel of Linear Transformation Consider the linear mapping F : Rª → Rª F(x, y, z, t) = (x −y+z+t, 2x - 2y+3z + 4t, 3x − 3y+4z + 5t) a. Find a basis and the dimension of the image of F. b. Find a basis and the dimension of hte kernel of the map of F.

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.

ProofProve in full detail that M2,2, with the standard operations, is a vector space.

Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}

Let T:R3R3 be the linear transformation that projects u onto v=(2,1,1). (a) Find the rank and nullity of T. (b) Find a basis for the kernel of T.

Finding the Nullity and Describing the Kernel and Range In Exercises 33-40, let T:R3R3be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. T is the projection onto the vector v=(1,2,2): T(x,y,z)=x+2y+2z9(1,2,2)

Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.

Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.

Linear Transformation Given by a MatrixIn Exercises 23-28, define the linear transformation T:RnRmby T(v)=Av. Use the matrix A to a determine the dimensions of Rnand Rm, b find the image of v, and c find the preimage of w. A=[100113], v=(3,5), w=(5,2,1)

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).