How is Lyddie independent throughout her life? What are the events that show she is independent?

Lyddie is independent because she takes charge and doesn’t let others tell her what to do.

Lyddie shows independence when she saves her family from the bear, when she gets her own job after being fired from the pub, and when she protects Brigid from Mr. Marsden.

Lyddie’s independence showed through in every challenge she faced.  The first time we see Lyddie challenged is when the bear attacks.  Lyddie and her family are in their house when the bear comes in.  Lyddie takes charge immediately and orders everyone into the loft.

They obeyed her, even Mama, though Lyddie could hear her sucking in her breath. Behind Lyddie's back, the ladder creaked, as two by two, first Charles and Agnes, then Mama and Rachel, climbed up into the loft. Lyddie glared straight into the bear's eyes, daring him to step forward into the cabin. (Ch. 1)

This incident demonstrates how Lyddie is able to think quickly and react even when no one else can.  Her mother should have been the one to protect the family, but she was incapable of doing so and Lyddie stepped up.  Not only was no one harmed, but the bear left because Lyddie did not back down.

When Lyddie is hired out to the tavern owner Mrs. Cutler, she is horrified.  She doesn't want to be anyone's slave.  When Lyddie is fired from this same job, she does not give up.  She takes control of her own life and goes to get a job in a factory.

"I'm going to be a factory girl, Triphena."
"You what?"
"I'm free. She's set me free. I can do anything I want. I can go to Lowell and make real money to pay off the debt so I can go home." (Ch. 6)

Lyddie still wants to work to pay off the family debts and hopefully get the family back together again someday, so she takes the factory job.  Since it was her own choice, she feels more independent.  She is a hard worker and demonstrates her worth to the factory, which gives her more and more work.

Part of Lyddie's job is to train Brigid, an Irish girl who comes to the factory knowing nothing. Lyddie is annoyed at first to have a trainee slowing her down, but she comes to be very protective of Brigid, teaching her to read and looking out for her.  When Brigid is attacked by Mr. Marsden, Lyddie intervenes.  She is fired for doing so.

Even though Lyddie has lost her job, she makes sure to continue protecting Brigid.  She writes a letter to Mr. Marsden's wife which she tells Brigid to mail if anything happens.

"It can't be helped. It's done. But they must not dismiss you. I've already written a letter to Mr. Marsden. I told him if he dismissed you or bothered you in any way I would tell his wife exactly what happened in the weaving room. Now here is the letter addressed to her. If there is any problem you must mail it at once." (Ch. 22)

Lyddie's reaction to being fired, again, demonstrates her independence and reliance.  Lyddie insists on helping Brigid and making sure that Brigid is okay.  Lyddie knows that she will be all right herself because she never lets anyone get her down.