Exercise 4.1.1

T[x_]:=Array[Sum[x[[j]],{j,6-#}]&,4]
U=Table[Subscripted[u[i]],{i,5}];
U1=Table[Subscripted[u1[i]],{i,5}];
U2=Table[Subscripted[u2[i]],{i,5}];
T[U]

a)

T[U1+U2]-T[U1]-T[U2]

T[k U]-k T[U]//Simplify

b)

e=IdentityMatrix[5]

Do[Print[T[e[[i]]]],{i,5}]

Exercise 4.1.2

U={{1,1,1,0},
   {1,1,0,0},
   {1,0,0,0}};

a)

NullSpace[U]

Clear[T]
<<LinearAlgebra`Master`
T[v_]:=v-Projection[v,NullSpace[U][[1]]]
V=Table[Subscripted[v[i]],{i,4}];
T[V]

b)

V1=Table[Subscripted[v1[i]],{i,4}];
V2=Table[Subscripted[v2[i]],{i,4}];
T[V1+V2]-T[V1]-T[V2]

T[k V]-k T[V]

c)

T[T[V]]-T[V]

d)

u=Sum[K[i]U[[i]],{i,3}]

T[u]-u

Exercise 4.1.3

M=Array[1/(#1+#2)&,{7,5}]; M//MatrixForm

Length[NullSpace[M]]

B=GramSchmidt[Transpose[M],Normalized->False]

Clear[T]
T[v_]:=Sum[Projection[v,B[[i]]],{i,5}]
v={1,2,3,4,5,6,7};
T[v]

Clear[v]
V=Table[Subscripted[v[i]],{i,7}];
V1=Table[Subscripted[v1[i]],{i,7}];
V2=Table[Subscripted[v2[i]],{i,7}];
T[V1+V2]-T[V1]-T[V2]//Expand

T[k V]-k T[V]//Expand

Exercise 4.1.4

Do[p[i]=(x-2)^(i-1),{i,6}]
B=Transpose[Table[Coefficient[p[i],x,j-1],{i,6},{j,6}]];
B//MatrixForm

a)

NullSpace[B]

b)

IB=Inverse[B]; IB//MatrixForm

Clear[T]
T[p_]:=IB.Table[Coefficient[p,x,i-1],{i,6}]
p0=Sum[Subscripted[a0[i-1]]x^(i-1),{i,6}];
p1=Sum[Subscripted[a1[i-1]]x^(i-1),{i,6}];
p2=Sum[Subscripted[a2[i-1]]x^(i-1),{i,6}];
T[p1+p2]-T[p1]-T[p2]//Simplify

T[k p0]-k T[p0]//Simplify

c)

Do[Print[T[x^i]],{i,5}]

d)

T[Sum[Subscripted[a[i-1]]x^(i-1),{i,6}]]

Exercise 4.1.5

Clear[u,U]
u[i_]:=Table[Subscripted[a[i,j]],{j,5}]
U[i_]:=Table[Subscripted[b[i,j]],{j,5}]
L[v_]:=Det[{u[1],u[2],u[3],u[4],v}]
L[U[1]+U[2]]-L[U[1]]-L[U[2]]

k L[U[0]]-L[k U[0]]//Expand