In　this　paper,　we　present　the　stability　and　error　analysis　of　two　fully　discrete　IMEXLDG　schemes,　combining　local　discontinuous　Galerkin　spatial　discretization　with　implicit-explicit　Runge--Kutta　temporal　discretization,　for　the　linearized　one-dimensional　KdV　equations.　The　energy　stability　analysis　begins　with　a　series　of　temporal　differences　about　stage　solutions.　Then　by　exploring　the　stability　mechanism　from　the　temporal　differences,　and　by　constructing　the　seminegative　definite　symmetric　form　related　to　the　discretization　of　the　dispersion　term,　and　by　adopting　the　important　relationships　between　the　auxiliary　variables　with　the　prime　variable　to　control　the　antidissipation　terms,　we　derive　the　unconditional　stability　for　a　discrete　energy　involving　the　prime　variable　and　all　the　auxiliary　variables,　in　the　sense　that　the　time　step　is　bounded　by　a　constant　that　is　independent　of　the　spatial　mesh　size.　We　also　propose　a　new　projection　technique　and　adopt　the　technique　of　summation　by　parts　in　the　time　direction　to　achieve　the　optimal　order　of　accuracy.　The　new　projection　technique　can　serve　as　an　analytical　tool　to　be　applied　to　general　odd　order　wave　equations.　Finally,　numerical　experiments　are　shown　to　test　the　stability　and　accuracy　of　the　considered　schemes.