Matrices

Rank of Matrices

Definition

\(A\in M_{m\times n}(F)\), \(L_a:F^n\rightarrow F^m, L_A(x)=Ax\)

\(\text{Rank}(A)=\text{Rank}(L_A)\)

Theorem

The rank of a matrix equals the number of linearly independent column vectors of A, i.e., it is the dimension of the span of columns of A.

Proof.

\(A\in M_{m\times n}(F)\)

\(\text{Rank}(A)=\text{dim}(R(L_A))\)

Let \(\beta\) be standard basis of \(F^n\)

\(R(L_A)=\text{span}(L_A(e_i))\)