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Let V be a vector space, and T∈L(V). Show that the following statements (i), (ii) are equivalent:

(i) There exists a direct sum decomposition V=W⊕Z into two subspace, with T the projection from V onto W along Z.

(ii) T∘T=T.

any one can help with it, i have no idea with this question

(i) $Longrightarrow$ (ii):

$V = W oplus Z; tag 1$

by definition means that

$V = W + Z, ; W cap Z = {0}; tag 2$

we note that the decomposition of any $v in V$ into

$v = w + z, ; w in W, ; z in Z, tag 3$

is unique, for if

$w_1 + z_1 = w_2 + z_2, tag 4$

then

$W ni w_1 - w_2 = z_2 - z_1 in Z; tag 5$

thus,

$w_1 - w_2 = 0 = z_2 - z_1 Longrightarrow w_1 = w_2, ; z_1 = z_2 tag 6$

as claimed; therefore, since $w$ is unambiguously determined by $v$, we may define a function

$T:V to W, ; T(v) = T(w + z) = w; tag 7$

we investigate the linearity of $T$: if

$v = av_1 + v_2, tag 8$

we may uniquely write

$v_1 = w_1 + z_1, ; v_2 = w_2 + z_2, tag 9$

whence

$v = a(w_1 + z_1) + w_2 + z_2 = (aw_1 + w_2) + (az_1 + z_2) in W + Z, tag{10}$

uniquely; it follows that

$Tv = T(av_1 + v_2) = aw_1 + w_2 = aTv_1 + Tv_2, tag{11}$

establishing the linearity of $T$.

We compute

$T^2(w + z) = T(T(w + z)) = Tw = T(w + z), tag{12}$

whence

$T^2 = T. tag{13}$

(ii) $Longrightarrow$ (i):

$T^2 = T Longrightarrow T(T - I) = T^2 - T = 0; tag{14}$

set

$W = T(V); tag{15}$

then, via (14):

$w in W Longrightarrow exists v in V, ; w = Tv; ; Tw = T^2v = Tv = w; tag{16}$

we see that $T$ fixes $w$ pointwise; it acts as the identity on the subspace $W$. We may also set

$Z = (I - T)V; tag{17}$

then, again by (14),

$z in Z Longrightarrow exists v in V, z = (I - T)v; ; Tz = T(I -T)v = 0 Longrightarrow z in ker T; tag{18}$

likewise,

$z in ker T Longrightarrow Tz = 0 Longrightarrow z = Iz - Tz = (I - T)z Longrightarrow z in (I - T)V = Z; tag{19}$

thus,

$Z = ker T; tag{20}$

now if

$y in Z cap W, tag{21}$

we have

$y = Tv, ; v in V; tag{22}$

$Ty = 0; tag{23}$

therefore, again invoking (14),

$y = Tv = T^2v = T(Tv) = Ty = 0; tag{24}$

we have then shown that

$Z cap W = {0}; tag{25}$

finally, for $v in V$,

$v = Iv - Tv + Tv = (I - T)v + Tv in Z + W; tag{26}$

(25) and (26) show that

$V = W oplus Z; tag{27}$

(16) and (20) show that $T$ is a projection onto $W$ "along $Z$". Thus we have (ii) $Longrightarrow$ (i).

The implication (i) $Rightarrow$ (ii) follows from the definition.

The implication (ii) $Rightarrow$ (i) is not hard to verify once you know that $W=operatorname{im}T$ and $Z=operatorname{ker}T$.