I think my problem is that I wasn't making the connection that I was doing row operations hence when I was working with a column space I need to transpose so the row operations don't result in a matrix that is not a linear combination of the original matrix. If that makes sense.
To Zondrina:
In regards to your answer, would it be ok then, anytime it asks to find a basis for a row space to only perform row operations to get the answer? And at the same time, whenever asked to find a basis for the column space, only perform column operations to get an answer? In the case...
In my linear algebra class we previously studied how to find a basis and I had no issues with that. Now we are studying the basis of a row space and basis of a column space and I'm struggling to understand the methods being used in the textbook. The textbook uses different methods to find these...