Linear Algebra Review for Test Tomorrow--Vector Spaces

I am going through a Review sheet for a test tomorrow morning and I came across this proof that I do not know how to do.  He gave us a list of proofs, at least two of which will be on the test.

 

3)  Let V and W be vector spaces in.

     Prove that VW = U SAY
is also a vector space.  You must give a reason for each step.


THE NECESSARY AND SUFFICIENT CONDITION FOR U TO BE A VECTOR  SPACE IS THAT IF A AND B ARE 2 VECTORS IN U , AND X AND Y ARE 2 SCALARS IN THE FIELD , THEN AX+BY IS ALSO A VECTOR IN  U.

SINCE A AND B ARE VECTORS IN  U , THEY MUST ALSO BE VECTORS IN  V AND W , SINCE V INTERSECTION W IS U.UNLESS A AND B ARE VECTORS IN BOTH V AND W, THEY CAN NOT BE PRESENT IN V INTERSECTION W=U.

NOW SINCE A AND B ARE VECTORS IN V FOR ANY 2 SCALARS X AND Y IN THE FIELD,AX+BY IS A VECTOR IN V AS V IS A VECTOR SPACE.

||| LY, SINCE A AND B ARE VECTORS IN W FOR ANY 2 SCALARS X AND Y IN THE FIELD,AX+BY IS A VECTOR IN W AS W IS A VECTOR SPACE.

HENCE AX+BY IS A VECTOR IN BOTH V AND W .
HENCE AX+BY IS A VECTOR IN V INTERSECTION W =U
THAT IS WE PROVED THAT
IF A AND B ARE 2 VECTORS IN U , AND X AND Y ARE 2 SCALARS IN THE FIELD , THEN AX+BY IS ALSO A VECTOR IN  U.
HENCE U IS A VECTOR SPACE AS PER THE THEOREM GIVEN AT THE BEGINING.