- #1

td21

Gold Member

- 178

- 8

## Homework Statement

Let U be a subspace of V. Suppose that U is a invariant subspace of V for every linear

transformation from V to V. Show that U=V.

## Homework Equations

no

## The Attempt at a Solution

Assume U is not trivial: Now we only need to show that U = V. Let dimV = n: We can choose

a nonzero vector u in U and then extend u to a basis u1,u2,u3,....u{n-1} for V.

There is a linear map k such that

k(u) = u1;

k(u1) = u2;

k(u2) = u3;

...

k(u{n-1})=u

As U is invariant, u1 also in U,then as k(u1)=u2,u2 also in U,.....................then u{n-1} also in U.

On the other hand, as u1,u2,u3,....u{n-1} is a basis for V, we have

span(u1,u2,u3,....u{n-1}) = V:

So it follows that U = V.

The difficulty is proving k is a linear transformation, and if it is, is the argument above correct? Thanks!