Question

let A be a 7*5 matrix with rank(A)=2 complete dim(row space of A) , dim( column space of A) ,dim (null space of A) and (null space of A^t)

Answer 1

dim(row space of A)

$dim (rowspace(A)) = rank(A) = dim (colspace(A))\\ dim (rowspace(A))+dim(null (A))=7\\ dim (rowspace(A))=7-dim(col (A))\\ dim (rowspace(A))=7-4=3\\$

dim( column space of A)

$[A]_{5*7}$ gives us a transformation T with domain v of dimension T

$dim(col (A))+dim(null (A))=7\\ dim(col (A))=7-dim(col (A))\\ dim(col (A))=7-4=3\\$

dim (null space of A)

$dim(null (A))+dim(col (A))=7\\ dim(null (A))=7-dim(col (A))\\ dim(null (A))=7-5=2\\$

(null space of A^t)

$(null space of A^t)= (null space of A)=2$