When we solve for inner product of $\rvert a \rangle \cdot \rvert b \rangle$ we solve for $\langle a \rvert b \rangle$ where $\langle a \rvert$ is complex conjugate of $\rvert a \rangle$. However this confuses me because in linear algebra, $u \cdot v$ is $uv^*$. The latter vector is conjugated. Why does braket notation conjugate prior vector and linear algebra conjugate latter vector?
Because it is the bra that is associated with dual vector, not the ket. If it were bra that was used as the element of original Hilbert space, then it would be like linear algebra. Blame Dirac for wanting operators to act from the left AND on the"flat side", just because it looks better than acting from the left and on the pointy side.