Linear Algebra and sums

[bry32321]

Let the vector space V=(F^3), U={(x,0,z) : x, z are elements of F}, W={(x, 0, x) : x is an element of F}.
Decide and justify : a.) if U+W=V and b.) if U+W= the direct sum of U and W


I can see why U+W is not a direct sum because a counter example of U and W being a direct sum is

(4,0,8)=(5,0,1)+(-1,0,7) and (4,0,8)=(6,0,2)+(-2,0,6)

but why for part a is U+W not equal to V?

is this because for the second component for the sum of U and W, it will always be 0?

 

[stapel]

Let the vector space V=(F^3), U={(x,0,z) : x, z are elements of F}, W={(x, 0, x) : x is an element of F}.
Decide and justify : a.) if U+W=V and b.) if U+W= the direct sum of U and W

I can see why U+W is not a direct sum because....

How is your book defining "+", that "U+W" does not stand for "the direct sum of U and W"?

but why for part a is U+W not equal to V?

Is it possible to create all elements of V by adding elements from U and W? For instance, can you create the element (0, 1, 0)?