Prove that the set of fixed points of a linear transformation T : V rightarrow V is a subspace of
V.
Solution
Dear William,
In order to show that a subset P of a vector space V is a subspace of that vector space, we must
show three things. First of all we must show that the 0 vector is contained in P. Secondly, we
must show that if v and w are in P, then v+w is in P also. Finally we must show that if v is in P
and c is a scalar, then cv is in P. The last two conditions are summarized by saying that P is
closed under the taking of linear combinations. If these three conditions are satisfied, then P will
be a subspace.
So now I begin the proof.
First of all, the zero vector 0 is a fixed point because all linear transformations map the zero
vector to the zero vector of the image space, which is the same as the domain space in this case.
Thus T(0) =0. Thus zero is a fixed point. As to the second part, assume that v and w are fixed
points. Then we wish to show that v+w is also a fixed point. By definition T(v) -> v and T(w)-
>w. Furthermore, since T is a linear transformation we have T(v+w) = T(v)+T(w) = v+w. Thus
v+w is a fixed point if v and w are. Finally, let us assume that v is a fixed point and that c is a
scalar. Then we wish to show that cv is also a fixed point. We have T(cv) = cT(v) = cv, since Tis
linear and v is a fixed point. Thus we have shown that the set of fixed points contains the zero
vector and is closed under the taking of linear combinations. Thus the set of fixed points of a
linear transformation T: V->V is a subspace of V.
I hope that this helps!!
David
V.
Solution
Dear William,
In order to show that a subset P of a vector space V is a subspace of that vector space, we must
show three things. First of all we must show that the 0 vector is contained in P. Secondly, we
must show that if v and w are in P, then v+w is in P also. Finally we must show that if v is in P
and c is a scalar, then cv is in P. The last two conditions are summarized by saying that P is
closed under the taking of linear combinations. If these three conditions are satisfied, then P will
be a subspace.
So now I begin the proof.
First of all, the zero vector 0 is a fixed point because all linear transformations map the zero
vector to the zero vector of the image space, which is the same as the domain space in this case.
Thus T(0) =0. Thus zero is a fixed point. As to the second part, assume that v and w are fixed
points. Then we wish to show that v+w is also a fixed point. By definition T(v) -> v and T(w)-
>w. Furthermore, since T is a linear transformation we have T(v+w) = T(v)+T(w) = v+w. Thus
v+w is a fixed point if v and w are. Finally, let us assume that v is a fixed point and that c is a
scalar. Then we wish to show that cv is also a fixed point. We have T(cv) = cT(v) = cv, since Tis
linear and v is a fixed point. Thus we have shown that the set of fixed points contains the zero
vector and is closed under the taking of linear combinations. Thus the set of fixed points of a
linear transformation T: V->V is a subspace of V.
I hope that this helps!!
David
