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## Homework Statement

Suppose that V is the direct sum U[itex]\oplus[/itex]U' where U, U' are subspaces of V, which is a subspace of F

^{n}. Define P:V→V as follows: if v[itex]\in[/itex]V then we know we can write v uniquely as v=u+u' for some u[itex]\in[/itex]U, u'[itex]\in[/itex]U'. Define P(v)=u. Show that:

a) P is linear

b) P

^{2}=P (a linear function with this property is called a projection).

Let P'=I-P where I is the identity function

c) PP'=0=P'P

d) U=KerP', U'=KerP

## The Attempt at a Solution

Since V is a direct sum of U and U', then U[itex]\bigcap[/itex]U'={

__0__}

To prove that P is linear, I need to prove that P(v+v')=P(v)+P(v') and P(cv)=cP(v)

[itex]P(v+v') = u[/itex]

[itex]P(v)+P(v') = u+u'[/itex]

Which obviously doesn't work. I'm using the assumption that v+v' is still in V, which is clearly an incorrect assumption. I also tried this:

[itex]P(v+v') = P((u_1+u_1')+(u_2+u_2')) = P(u_1'+u_2'+u_1+u_2)[/itex]

[itex]P(v)+P(v') = P(u_1+u_1')+P(u_2+u_2')[/itex]

For closed under multiplication, I didn't even know where to start.

Sorry for not showing very much work, but I'm so stuck that there is no work to show... Thanks!