Suppose V is finite dimensional U is a subspace of V and T U

Suppose V is finite dimensional, U is a subspace of V, and T: U rightarrow W is a linear transformation. Prove that there is a linear transformation. 5: V rightarrow W such that S(u) = T(u) whenever u U. (another way to say this is S|(u = T i.e. \"5 restricted to U is T. Find a linear transformation T: F3 -* F3 with T(1,1,0) = (1,0,0) and T(l, 0,1) = (0.1.0) (that Ls, find an explicit formula for T(x, y, z).

Solution

a)

Define S(v)=0 for all v not in V but not in U

And S(u)=T(u) for all u in U

Hnece S is the required linear transformation

b)

Since F3 is a 3 dimensional space we need to find action of T on three linearly independent vectors to be able to define T

So, let us consider action on: v3= (0,1,-1)

Denote (1,1,0)=v1,(1,0,1)=v2

T(0,1,1)=(0,1,0)

e2=(1,1,0)/2-(1,0,1)/2+(0,1,1)/2=(0,1,0)=(v1-v2+v3)/2

e1=v1-(v1-v2+v3)/2=(v1+v2-v3)/2

e3=v3-e1=(-v1-v2+3v3)/2

T(x,y,z)=xT(e1)+yT(e2)+zT(e3)=0.5xT(v1+v2-v3)+0.5yT(v1-v2+v3)+0.5zT(-v1-v2+3v3)