Stabilized tangent bundle of real projective space ($T{\mathbb{RP}}^m\oplus {\epsilon }^1\cong (m+1)\gamma$)

For the real tautological bundle $\gamma$ over Real projective space, the tangent space at a line is $\mathrm{Hom}(\gamma ,{\gamma }^{\perp })$. Adding $\mathrm{Hom}(\gamma ,\gamma )\cong {\epsilon }^1$ gives $\mathrm{Hom}(\gamma ,{\epsilon }^{m+1})\cong (m+1)\gamma$, using a metric to identify a real line with its dual. Hence its total Stiefel–Whitney class is $(1+a{)}^{m+1}$, with $a=w_1(\gamma )$.