direct sum vector space

Last problem:
Suppose that V is a vector space over any field F and that U and W are
subspaces of V.
Let V=C[X], the vector space of all polynomials over complex numbers. Let U=Span{1+(1+i)X^2, (1+2i)+iX+X^4} and W=Span{X+(2-i)X^2+(3+i)X^3, (2+2i)+(1+i)X+3X^2+(3+i)X^3+2iX^4}. Show that the sum U+W is actually a direct sum and conclude that U+W has a basis with four elements.

Thank you so much.

Quote:

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Originally Posted by USCGuy

Last problem:
Suppose that V is a vector space over any field F and that U and W are
subspaces of V.
Let V=C[X], the vector space of all polynomials over complex numbers. Let U=Span{1+(1+i)X^2, (1+2i)+iX+X^4} and W=Span{X+(2-i)X^2+(3+i)X^3, (2+2i)+(1+i)X+3X^2+(3+i)X^3+2iX^4}. Show that the sum U+W is actually a direct sum and conclude that U+W has a basis with four elements.

Thank you so much.

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Perhaps, the vectors over the complex field,
$\displaystyle (1,0,1+i,0,0),(1+2i,i,0,0,1),(0,1,2-i,3+i,0),(2+2i,1+i,3,3+i,2i)$
Are linearly independent.
These, vectors correspond to the corresponding coefficients in the polynomials.

Huh?

Could you explain it in a little more detail?