Rank

Prerequisites

• matrix

Linear combination

A linear combination of a sequence of vectors $v_1,v_2,\cdots ,v_k$ is a vector of the form $c_1{v}_1+c_2{v}_2+\cdots +c_k{v}_k$, where $c_i\in F,v_i\in F^n$.

Linear Independence

A sequence of vectors $v_1,v_2,\cdots ,v_k$ is said to be linearly independent if the only solution to the equation $c_1{v}_1+c_2{v}_2+\cdots +c_k{v}_k=0$ is the trivial solution $c_1=c_2=\cdots =c_k=0$. Where $c_i\in F,v_i\in F^n$.

Span

The span of a sequence of vectors $v_1,v_2,\cdots ,v_k$ is the set of all linear combinations of $v_1,v_2,\cdots ,v_k$, denoted by $span(v_1,v_2,\cdots ,v_k)$.

$span(v_1,v_2,\cdots ,v_k)=\{c_1{v}_1+c_2{v}_2+\cdots +c_k{v}_k|c_i\in F,v_i\in F^n\}$

$span(v_1,v_2,\cdots ,v_k)$ is a subspace of $F^n$.

If $A=(v_1,v_2,\cdots ,v_k)\in F^{n\times k}$, then $col(A):=span(A):=span(v_1,v_2,\cdots ,v_k)$.

Elementary transformations

• Elementary row operations
  • swap row $i$ and row $j$
  • multiply row $i$ by a nonzero scalar $c$
  • add $c$ times row $i$ to row $j$
• Elementary column operations
  • swap column $i$ and column $j$
  • multiply column $i$ by a nonzero scalar $c$
  • add $c$ times column $i$ to column $j$

Elementary row operations can be implemented by multiplying $A$ by an elementary matrix from the left.

Elementary column operations can be implemented by multiplying $A$ by an elementary matrix from the right.

Elementary row operations not change the dimension of the row space and the column space.

Elementary column operations not change the dimension of the column space and the row space.

Elementary row operations can be used to reduce a matrix to its reduced row echelon form. Dimension of the row space equals Dimension of the column space in reduced row echelon form.

Therefore, $\dim (col(A))=\dim (row(A))$.

Rank

$A\in F^{m\times n}$ is a matrix.

$rank(A)=\dim (col(A))=\dim (row(A))$