freshman student here looking to clarify the concepts.
1.) Is the column space also the solution space of a matrix?
2.) Lets say a matrix is formed by a1,a2,a3,a4,a5. I find the rref form of this matrix; turns out a1,a2,a3 are linearly independent. Can it suffice to say that a1,a2,a3 are the basis for both solution space and column space of a matrix? Can this tell me anything about row space?
3.) What is the purpose of the row space?
4.) Is the basis for row space also the basis for the solution space in a square matrix? Here on out it becomes increasingly difficult for me to visualize.
5.) Apart from dimension theorem, what other purpose does rank actually serve? So I am familiar with the formulas regarding rank; rank <= min(n,m) or rank = min(dimension(rowspace),dimension(colspace)), but I am not actually sure I understand the implications of this.
6.) If dim(rowspace)<dim(colspace), can this tell me anything about the solution space or any implications (apart from implications on the rank)?
Sorry for the long-winded question. Hope anyone can provide insight on anything above, for me and my friends.
Best Answer
Let's go through these one by one.
$$\text{rank }A=\text{dim Col }A=\text{dim Row }A$$