= Solution
If Alice's <basic measurement> returns a normalized vector $|u\rangle_A$, form Bob's unnormalized conditional vector
$$
|\widetilde\chi_u\rangle_B=(\langle u|_A\otimes I_B)|\chi\rangle_{AB}.
$$
The <Born rule> and normalization give
$$
\boxed{p_u=\|\widetilde\chi_u\|^2,\qquad
|\chi_u\rangle_B=\frac{|\widetilde\chi_u\rangle_B}{\sqrt{p_u}}\quad(p_u>0).}
$$
The joint <post-measurement state> is $|u\rangle_A\otimes|\chi_u\rangle_B$. For $|u\rangle_A=\alpha|0\rangle_A+\beta|1\rangle_A$ with real coefficients and $\alpha^2+\beta^2=1$, contraction of the <Bell state> gives
$$
(\langle u|\otimes I)|\phi_+\rangle
=\frac{\alpha|0\rangle+\beta|1\rangle}{\sqrt2},
\qquad p_u=\frac12.
$$
Thus \b[Bob's normalized conditional state is $\alpha|0\rangle_B+\beta|1\rangle_B$]. For complex coefficients the contraction instead contains their <complex conjugates>; this is the same conjugation underlying <conjugate-basis steering of a Bell pair>.
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