Linear Algebra Done Right · Problem 6C.21

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · pseudoinverse

Problem

Suppose T∈L(F3,F2)T \in \mathcal{L}(\mathbf{F}^3, \mathbf{F}^2) is defined by T(a,b,c)=(a+b+c,2b+3c).T(a, b, c) = (a + b + c, 2b + 3c). (a) For (x,y)∈F2(x, y) \in \mathbf{F}^2, find a formula for T†(x,y)T^\dagger(x, y). (b) Verify that the equation TT†=Prange⁡TTT^\dagger = P_{\operatorname{range} T} from 6.69(b) holds with the formula for T†T^\dagger obtained in (a). (c) Verify that the equation T†T=P(null⁡T)⊥T^\dagger T = P_{(\operatorname{null} T)^\perp} from 6.69(c) holds with the formula for T†T^\dagger obtained in (a).

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