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I am trying to get a head start on Linear Algebra before i start taking classes in a couple weeks. I am about to go into my second year undergraduate and all i have behind my belt is calculus (single varialbe, multivariable, and vector analysis (curl, divergence, etc)).

I am having a ton of trouble wrapping my head around stuff like vector spaces. Spans are especially difficult for me to understand because i can't visualize them. I understand what is said when i read the book, but it doesn't help me with go further because of the way it's presented.

I am using Axler's Linear Algebra Done Right to study from and here is an excerpt from chapter 2 about the lemme i can't understand.

It is used for this proof

How can I adjust my way of thinking as to not get bogged down by the the strange new world of linear algebra? It has new notation, new ideas, and new things that i simply can't understand!!!