10-6 LINEAR EQUATIONS 735
K. We can use these vectors as column vectors to form an n by k matrix D:
D = l " l i= ", i= 1,...,k
Now the vectors of the linear variety L are all vectors of the form
x c+t11 + t22,+-... + tkk
where tI,...., tk are arbitrary scalars or, in terms of coordinates, are all
(x1..... x) such that
x1 = c1 + d11t1 + d12t2 +... + dlktk
(10-63)
xn ~ = c+ dnitl + dn2t2 + " + d ktk
Thus
x = c + Dt (10-64)
where t is an arbitrary column vector in V. All vectors in (10-64) are written
vertically. E(quations (10-63) (or (10-64)) are termed a parametric representation
of the linear variety L in terms of parameters t1,..., tk. (When k = 1 and
n = 2, we obtain the parametric equations of a line, as in Section 1-15.)
When c = 0, L K and the linear variety is a subspace of V". Thus
x = Dt
is the parametric representation of a k-dimensional subspace K in terms of
parameters t1,...., tk. When k = n, t is an arbitrary vector of V? and the
equation x= Dt is the same as x= t161 + -... + t, the representation
of x in terms of a basis 61...., n"
For general k, the equation x = Dt can be regarded as a linear mapping
D of \ 1 into Vn whose range is K. Because c,..., k are a basis for K,
the representation of x as )t is unique; that is, D is a one-to-one mapping
as in No. 5 above, with D of maximum rank, here equal to k).
We remark finally that, given a subspace Z of V1 of dimension k, there
are systems of n - k linear homogeneous equations in n unknowns whose
solutions are the vectors in Z. Furthermore, if L is a k-dimensional linear
variety of V, not a subspace of V,, then L is the set of solutions of n - k
linear equations: a:x1 + a + -... ainxn = b, i 1,2...., n- k, where
the matrix (a;) has rank n - k. These assertions follow from Theorems 1 and
2 in Section 10-4.
Results Formulated Without Vectors. Because of their importance for
many applications, we restate some of the results for the case m = n in other
language:
THEOREM 3. Let there be given the n equations in n unknowns
(1 1X ti x... + a n n -- x, bl..... (nl, 1 " x +- "' - annn x, bn (10-65)
with co esonding homogeneous equations
a11x1 + --... + alxn = 0,... anx1 +... + annxn = 0 (10-66)
