2.4 The four fundamental subspaces 25Definition 2.3.17 In any system Ax = b <£> Ux = c, we can partition the
unknowns Xi as basic (dependent) variables those that correspond to a column
with a nonzero pivot 0, and free (nonbasic,independent) variables correspond-
ing to columns without pivots.
We can state all the possible cases for Ax = b as we did in the previous
remark without any proof.
Theorem 2.3.18 Suppose the m by n matrix A is reduced by elementary row
operations and row exchanges to a matrix U in echelon form. Let there be r
nonzero pivots; the last m — r rows of U are zero. Then, there will be r basic
variables and n — r free variables as independent parameters. The null space,
Af(A), composed of the solutions to Ax = 8, has n — r free variables.
If n — r, then null space contains only x = 6.
Solutions exist for every b if and only if r = m (U has no zero rows), and
Ux = c can be solved by back-substitution.
If r < m, U will have m — r zero rows. If one particular solution x to
the first r equations of Ux = c (hence to Ax = b) exists, then x + ax, \/x G
Af(A) \ {6} , Va S R is also a solution.Definition 2.3.19 The number r is called the rank of A.2.4 The four fundamental subspaces
Remark 2.4.1 If we rearrange the columns of A so that all basic columns
containing pivots are listed first, we will have the following partition of U:A = [B\N] -> U =
UB\UN
o
-^v =Ir\VN
o
where B € Rm*r, N € M™x("-'-)j \jB <= Rrxr^ Uff £ Rrx(n-r)> o is an
(m-r) x n matrix of zeros, VN £ Krx(n-r>, and Ir is the identity matrix of
order r. UB is upper-triangular, thus non-singular.
If we continue from U and use elementary row operations to obtain Ir in
the UB part, like in the Gauss-Jordan method, we will arrive at the reduced
row echelon form V.2.4.1 The row space of A
Definition 2.4.2 The row space of A is the space spanned by rows of A. It
is denoted by 1Z(AT).
Tl(AT) = Spandat}^) =lyeRm:y = f>a< j= {d G Rm : 3y € Rm 9 yTA = dT).