Kunze Linear Algebra · Problem 5.3.9

The key insight is recognizing that permuting coordinates is fundamentally a rearrangement—it's reversible by construction. To prove $T$ is invertible, you need to show it's linear (respects addition and scalar multiplication) and bijective (one-to-one and onto). Linearity follows directly from how permutations distribute over coordinate operations. Bijectivity comes from the fact that $\sigma$ itself is a permutation, so applying the inverse permutation $\sigma^{-1}$ recovers the original vector—this inverse map is your explicit inverse operator.

The key insight is recognizing that permuting coordinates is fundamentally a rearrangement—it's reversible by construction. To prove $T$ is invertible, you need to show it's linear (respects addition and scalar multiplication) and bijective (one-to-one and onto). Linearity follows directly from how permutations distribute over coordinate operations. Bijectivity comes from the fact that $\sigma$ itself is a permutation, so applying the inverse permutation $\sigma^{-1}$ recovers the original vector—this inverse map is your explicit inverse operator.