In Reply to: Homework 1 - Q7 posted by Doug on October 08, 2002 at 23:14:00:
>Hi Professor Tao,
>I am kinda confused by Q7 (a) and (b). For (a), I try to let x1, x2 be contined in V such that x1 can be written as y1 + y2; x2 can be written as z1 + z2, in which y1, z1 are contined in W1 and y2, z2 are contained in W2. I then proceed to prove the validity of the zero element, vector addition and scalar multiplication. However, I don't understand the difference between (a) and (b). Could you give me some hints at solving (b)? Thanks!
>Best,
>Doug
Dear Doug,
Your approach to (a) is basically correct. For (b), it may
be clearer if we rephrase the question as follows:
23(b). Let W be a subspace of V which contains both W_1 and W_2.
Prove that W must also contain W_1+W_2.
(i.e. I have given the "any subspace" in Q23(b) a name - W.
A general rule of thumb in mathematics is that when things become
confusing, give distinct names to as many objects as possible).
So, your task is now:
* Given a vector space V;
* Given subspaces W_1, W_2, W of V;
* Given that W contains W_1;
* Given that W contains W_2;
-----------------------------
* Show that W contains W_1 + W_2.
This has some similarities with (a) but is asking a different question.
An example might help: suppose V = R^3, W_1 is the x-axis, and
W_2 is the y-axis. Then W_1 + W_2 is the xy-plane (why?).
Part (a) is then asking that the xy plane is a vector space
(closed under addition and scalar multiplication). Part (b)
is asking that any other subspace which contains the x-axis
and the y-axis, must automatically also contain the xy-plane.
Terry