... basis and adjoining gives this matrix.RepB,B( d dx) =1 00 2B,B(c) The images of the members of the domain’s basis are 1 d/ dx−→ 0, x d/ dx−→ 1, ex d/ dx−→ ex, and xex d/ dx−→ex+ ... is symmetricbecause hi ,j + gi ,j = h j, i+ g j, i(since hi ,j = h j, iand gi ,j = g j, i). Thus the subset is nonempty andclosed under the inherited operations, and so it is a subspace.Three.IV.1.16 ... h(v) isn j= 1hk ,j v j and so the i-th component of g ◦ h (v) is thisrk=1gi,kn j= 1hk ,j v j =rk=1n j= 1gi,khk ,j v j =rk=1n j= 1(gi,khk ,j )v j =n j= 1rk=1(gi,khk ,j )v j =n j= 1(rk=1gi,khk ,j )...