Prove The range of πk − πk+1 is the same as the range of (πk − πk+1)(Ik − Ik+1). More specifically, it coincides with the range of (I − Qa k+1 )(Ik − Ik+1). Also, any basis {ϕi} of the space Range(Ik − Ik+1) provides a basis of Range(πk − πk+1) defined by {(I − Qa k+1 )ϕi}.

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