The discussion revolves around determining the rank of a matrix, specifically addressing the relationship between linearly independent columns and rows. Participants explore the implications of counting both rows and columns in assessing rank.
The discussion is ongoing, with participants providing guidance on proving linear independence and clarifying concepts. There is an exploration of different interpretations regarding the rank of the matrix and the definitions involved.
Participants note the importance of understanding that the number of linearly independent rows cannot exceed the number of linearly independent columns, highlighting a fundamental theorem of linear algebra. There is also mention of potential confusion in terminology related to linear independence and dependence.
influx said:Homework Statement
Homework Equations
N/A
The Attempt at a Solution
I know that they got a rank of 2 since there are 2 linearly independent columns but what if we decided to count rows? In that case we would have 4 linearly independent rows which would suggest the rank is 4? How do we reconcile between these?
fresh_42 said:
No, I meant linear independency. My goal was to let the OP see by himself why this won't be possible and learn out of it.Ray Vickson said:
fresh_42 said:
He claimed the rows to be linear independent, so to ask for a proof seemed to be logic. Thus he would have learned multiple things: don't claim, what you cannot prove, the calculations with linear dependencies and linear independencies are basically the same, and a (failed) proof would have simultaneously shown two free parameters solving the dimension formula. All within this tiny example.Ray Vickson said: