# linear maps, rank

• Jan 27th 2011, 12:49 PM
rayman
linear maps, rank
Let $V=V(m,\mathbb{R})$ and $W=W(n,\mathbb{R})$ and let f be a matrix corresponding to a linear map from V to W. Verify that $rankf=rankf^{t}=rank(Mf^{t}N)$. Gdzie $M\in GL(m,\mathbb{R})$ oraz $N\in GL(n,\mathbb{R})$
• Jan 27th 2011, 01:16 PM
FernandoRevilla
One way:

$M(f^t)=[M(f)]^t$

where:

$M(f)$ : matrix of $f$ with respect to the basis $B_V$ and $B_W$

$M(f^t)$ : matrix of $f^t$ with respect to the dual basis $B_W^*$ and $B_V^*$

Fernando Revilla