Let V be a finite-dimensional inner product space. A linear operator U on V is called a partial isometry if there exists a subspace W of V such that ||U(x)|| = ||x|| for all x EW and U(x) = 0 for all x € W¹. Observe that W need not be U-invariant. Suppose that U is such an operator and {V₁, V2, ..., Uk} is an orthonormal basis for W. Prove the following results. (b) {U(v₁), U(v₂),..., U(Uk)} is an orthonormal basis for R(U).

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