Confused on definition of projection

[JonnyG]

My textbook says: "if ## V = W_1 \oplus W_2 ##,, then a linear operator ## T ## on ##V ## is the projection on ##W_1## along ##W_2## if, whenever ## x = x_1 + x_2##, with ##x_1 \in W_1## and ##x_2 \in W_2##, we have ##T(x) = x_1##"

It then goes on to say that "##T## is a projection if and only ##T^2 = T##.

But what if ##T = I## (the identity operator)? Then suppose ##V## is finite dimensional and ##W## is a subspace of ##V##. Then ##V = W \oplus W^{\perp}## so that any ##x \in V## has the form ##x = x_1 + x_2##. So by the first definition, ##I## is not a projection because ##I(x) = x = x_1 + x_2 \neq x_1 ##. But by the second definition, ##I## is a projection, because ##I^2 = I##.

What's going on here?

 

[Simon Bridge]

Are you sure the second statement is a definition ... could it be a consequence of applying the definition in a particular situation.

 

[JonnyG]

The exact wording in the book is "In fact, it can be shown (see Exercise 17 of Section 2.3) that ##T## is a projection if and only ##T^2 = T##. The article on Wikipedia also gives this definition.

 

[mathwonk]

Id is a trivial projection, based on the trivial decomposition of V into the sum of V and the zero subspace. note: Vperp = {0}. I.e. x1+x2 = x1 precisely when x2 =0.