### Range and Column Space

##### 2. Column Space

For a linear transformation T: RnRm given by an m by n matrix A = [a1 a2 ... an], we have (see this exercise for a more concrete example)

 T(x) = [ a11x1 + a12x2 + ... + a1nxn ] a21x1 + a22x2 + ... + a2nxn : am1x1 + am2x2 + ... + amnxn = x1[ a11 ] + x2[ a12 ] + ... + xn[ a1n ] a21 a22 a2n : : : am1 am2 amn = x1a1 + x2a2 + ... + xnan

Thus rangeT is given by all the linear combinations (i.e., for all choices of the coefficients x1, x2, ..., xn) of the columns of A.

The column space of a matrix A is

colA = {all the linear combinations of columns of A}
= {Ax: any xRn}
= {bRn: Ax = b has solutions}.

We remark that the column space of an m by n matrix is a subspace of Rm.

Example Because

(a + b, 3a - b, 2a + b) = a(1, 3, 2) + b(1, -1, 1),

the subspace

{(a + b, 3a - b, 2a + b): a, bR} ⊂ R3

given in an earlier example is in fact the columns space of the matrix

 A = [ 1 1 ]. 3 -1 2 1

Notice that the linear transformation in the earlier example is also given by the matrix.

We also note that since the system Ax = b has solutions for b = (3, 1, 4) and has no solution for b = (1, 0, 1), the vector (3, 1, 4) is in colA, and the vector (1, 0, 1) is not in colA.