User:IssaRice/Linear algebra/Rank

$T:V\to W$

$A$ is m by n matrix

• $\dim \operatorname {range} T$
• $\dim \operatorname {range} {\mathcal {M}}(T)$ (what basis?)
• dimension of row space of $A$
• dimension of column space of $A$
• number of non-zero rows in the echelon form matrix of $A$
• number of pivots in the echelon form matrix of $A$
• number of linearly independent rows in $A$
• number of linearly independent columns in $A$

I think the rank is determined by the matrix (i.e. you don't need to know the linear map). This means that if you take two different linear maps and can find some bases so that the matrix is the same for both, then the two maps must have the same rank.

The rank is also determined by the linear map (i.e. you don't need to know the matrix). This means that if you take two different bases and compute the matrix of the map with respect to both of them, the rank will be the same.