Question

Suppose U={(x, x, y, y) ∈ F⁴ :x, y ∈ F}. Find a subspace W of F⁴=U ∅ W

Answer 1

U={$(x,x,y,y)\in F_4:x,y\in F$ }

 =span {$(1,1,0,0),(0,0,1,1)$ }

For $W=span$ {$(1,0,0,0),(0,0,0,1)$ }

   

   ={$(x,0,0,y)\in F_4:x,y\in F$ }

Let $(a,b,c,d)\in U\cap W$

$\Rightarrow (a,b,c,d)\in U and (a,b,c,d)\in W$

$(a,b,c,d)\in U \Rightarrow a=b \text{ and } c=d~~~~~~~~-(1)\\ (a,b,c,d)\in W\Rightarrow b=0 \text{ and } c=0~~~~~~~~-(2)$

from (1) and (2) we have a=b=c=d=0

$U\cap W={0}~~~~~~~-(3)$

Let $(a,b,c,d)\in F_4$

$(a,b,c,d)=(b,b,c,c)+(a-b,0,0,d-c)$

Let $(a,b,c,d)\in F_4$

$(a,b,c,d)=(b,b,c,c)+(a-b,0,0,d-c)$

$\Rightarrow F_4=U+W~~~~~~~~~~-(4)$

From (3) and (4) we have-

$F_4=U \bigoplus W$