I'm studying elementary linear algebra right now, and the current section is on linear independence. As I create matrices from the vectors and row reduce them in a calculator, I get various results.
Some row reduced forms give nonzero integers on the right hand side except the bottom entry, which I understand to be linear dependence. Other forms have all zeros on the right hand side, which I understand to be linear independence. Another instance is when the right hand side is all zeros except the one on the bottom. Normally, I know that this means no solution, but my professor has stated, "The vector is not in the span of the other vectors."
I want to know the mathematical difference between linear independence and "not being in the span." If the vector is not in the span, does that mean it's not linearly dependent? If it cannot be linearly dependent, why not just call it linearly independent since the linear combination will never equal 0 unless every coefficient is 0?
If there are any errors in my understanding or assumptions, please correct me and shed light on my ignorant mind as the final exam approaches.
Another question is, can linearly independent vectors be written as a linear combination? I thought that the coefficients cannot all be 0, but my textbook seems to think they can be. It claims that the zero vector is a linear combination of two vectors, its linear combination that can never be 0 unless every coefficient is 0.
Sorry for another edit. The questions are just piling up. My textbook says that a zero vector is a linear combination of some vectors, but another nonzero vector is not a linear combination. Why can't every coefficient be 0 to make the nonzero vector a linear combination as well?