Basis of column space pivot columns corresponding to original matrix

Dimension r rank = no of pivots

Basis of Null Space - R (reduced row echolon form) [I F, 0 0] [-F ,I] I corresponds to the no of free variables. (special Solutions of Ux=0 or Rx=0)

Dimension = n-r

Basis of row space - pivoted rows of R matrix or first r rows of R(row space doesn’t change during gaussian elimination)

Dimension = r

Basis of Null space of A^t - Keep track of elementary transformations used to reduce the matrix to reduced row echolon form and build the final matrix E such that EA=R

the rows corresponding to the zero rows of R in E give the linear combination of rows of A for which we get the Zero rows making those rows solutions to the equation A^tx=0

those rows of E are the basis vectors of the null space of At

Dimension =m-r

NEW VECTOR SPACES CONSIDER MATRICES AS VECTORS!!!!!!!