The woman walked into a downtown shelter on Sept. 5 claiming she could only remember her first name.
"She has absolutely no recollection about her life," Const. Tony Vella told CBC News, adding that Linda was not carrying identification with which to confirm her name.
Police describe her as a white woman with blue eyes and straight blonde shoulder-length hair. She stands 5 feet 7 inches tall, weighs 150 pounds and is believed to be between the ages of 55 to 60 years old.
Linda showed up at the shelter wearing a dark blue jean jacket, black Capri−style pants, black slip−on shoes, a pink bandana, white ankle socks, glasses with light brown frames and a pink watch.
"It's a unique case. It doesn't occur all the time," said Vella, adding that they are sharing her photo to broaden their search.
May be 'functional amnesia'
Amnesia expert Shayna Rosenbaum, who works for York University and the Baycrest Centre for Geriatric Care, said she's only ever seen two cases like Linda's.
Rosenbaum suspects the woman has functional amnesia, likely caused by a traumatic event, which does not result from an injury to the brain.
"It could be that she might have lost a loved one, she might have undergone some other major stresser, perhaps loss of job," she told CBC's Nil Köksal.
Sometimes people with functional amnesia take months or years to recover — and in some cases they never do, said Rosenbaum.
Also facing uncertainty are the Toronto police, who are tracing the origin of Linda's clothing and checking the prescription of her eyeglasses, among other efforts to turn up clues.
"Our greatest resource right now is the media and the public," said Vella, urging anyone with tips to contact the police.
"Someone out there must know who she is."