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##"(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# I Confused on definition of projection

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