• sam0617
In summary, the conversation revolved around finding the dimension of a basis and the role of the original matrix A in this process. The individual asking the question was unsure of how to proceed and sought clarification on whether to first check for a basis and then count the non-zero rows of A, or to reduce A to REF or RREF before counting the non-zero rows. The conversation also touched on the concept of matrix rank and different methods for determining it.

#### sam0617

I tried googling this first and of course looked at my notes but the questions asked pertaining to dimensions were confusing me more so I thought I'd ask directly.

Sorry for asking the same question I'm sure someone else has asked.

Okay. So, to find the dimension, do I first make sure it's a basis ( linearly independent & spans )
then find how many non-zero rows the original matrix A had or
do I reduce the original matrix A to REF or RREF after I made sure it's a basis THEN count how many non-zero rows it has?

Thank you for the help.

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what are you trying to find the basis of, and what does A have to do with this?

I also don't understand what you're asking.

sam0617 said:
Okay. So, to find the dimension, do I first make sure it's a basis ( linearly independent & spans )
Find the dimension of what? Make sure that what is a basis?

sam0617 said:
then find how many non-zero rows the original matrix A had
Original matrix? Now I really don't understand.

sam0617 said:
do I reduce the original matrix A to REF or RREF after I made sure it's a basis THEN count how many non-zero rows it has?
A matrix can never be a basis. Are you asking how to determine if the set of rows of a given n×m matrix is a basis of the space of 1×m matrices?

Your question is vague, but it sounds like you just want to find the rank of the matrix. The rank is equal to the dimensionality of the linear subspace spanned by the columns (or rows) of the matrix.

I'm sure there are plenty of ways of determining matrix rank. (The wikipedia article on matrix rank suggests something about converting to row-echelon form)... but another way could be to take the trace of the projection of the matrix.

Thank you. Sorry for the confusion everyone.