Detectives said the breakthrough came after the family of Fatima Kama, 28, alerted relatives in Britain to say she had not arrived in Montreal. They in turn contacted the Metropolitan Police.
Ms Kama, in her early twenties, is understood to have been Lebanese although she may have held a Canadian passport. She had been living in the Marble Arch area of central London since the start of the summer. She had also visited Lebanon, returning to London 10 days ago.
Last night, detectives said that Ms Kama had last been seen alive at 2pm on Saturday, about nine hours before her body was discovered inside a suitcase in the terminal three short-stay car park at the airport west of London. She had been stabbed more than 10 times.
Police said Ms Kama was last seen with friends but they had been unable to trace her movements since then.
Obtaining a positive identification - which was made yesterday afternoon by one of her relatives - is a major breakthrough for police. Officers had stressed that until they could identify the victim it was virtually impossible to try to identify a possible motive.
"A family member saw the coverage in the media and put two and two together to make four," said Detective Chief Inspector Richard Taber, who is heading the inquiry.
"It is appalling that anyone should be stabbed and murdered but to have your loved one bundled ignominiously into a suitcase and callously abandoned in a car park, I think, is awful," he said, adding: "One of the theories we are working on at the moment is that whoever murdered this woman left the suitcase at the car park, then caught a flight out."
Detectives believe that their best chance in tracing the killer is through the closed-circuit television (CCTV) footage taken of all cars entering and leaving the car park. Up to 6,000 cars a day use the car park but officers are concentrating their efforts on tracing those who used it between 5pm and midnight on Saturday. Police believe they will need to speak to 1,500 drivers, who they will trace through the vehicle registration numbers which can be seen on the CCTV film footage.
Police are convinced that the suitcase was brought to the car park in a vehicle rather than by the Underground because of its weight. Ms Kama, who was 5ft, weighed around 10 stone. Three women had noticed the large black suitcase on Saturday evening and alerted staff. Detectives believed it had not been left long because security staff at the airport, which is a potential terrorist target, check the area every couple of hours.Reuse content