9-11 LINEAR MAPPINGS 681
(b) Determine the dimension of the space Z of continuous flnctions on )0 < x < v
which have 0 and as zeros.
9. Let au1t.... alm,...., ap be real numbers. Let W be those vectors (x1,...., x,)
of V, such that ax11x +.,. + al, xn = 0..., al1x1 +. ~. +,m,x, = 0.
(a) Show that W is a subspace of V,.
(b) Show that dim W'>- n - m.
(e) Give an example where i = 2 and dim W = n - 1 > n - m.
10. (a) Let W be the set of vectors from V having first coordinate 0. Find a basis
for V4 which contains no vector from W
(b) If U is a nontrivial subspace of a vector space V, show that there is a basis
for V containing rno vector from U. Contrast this result with Theorem 12.
11. Show that the following are linear varieties of V, and determine their dimension:
a) All (x1, x2, x, x4) such that x1 + x,2 + x3 + x4 = 1.
0h All(x1,, x3, x4)suchthatxl - x2 + x3 - =x4 2 x1 + 3x2 - x3 + x4 = 6.
() All (x, 2, 3, x,4) satisfying the conditions in both (a) and (b).
12. Prove: if U and W are subspaces of a vector space V then dim (U fl W) <
oin (dim U din W), with equality holding true only if the subspace of smaller
dimension is a sulbspa e of the other.
13. Let 1 and 1M be line ar varieties of a finite dimensional vector space V Show
that if L fl M is not empty and neither contains the other, then
dim (fl1 M) < min (dimn L, dim M)
14. L et V be an n- diensional space with basis {u1... u }. Let v- v* be the
correspondetnce specified in Theorem 11. Prove:
(a) u,. u} is a basis for 1.
fb) If {v1.... }vh is a linearly independent set of I then {1.. v} 4is a
linearly indlependent subset of V, and, conversely, if {v1,.., v 4 is a linearly
iidependent subset of V, then {v1... Vh } is a linearly independent subset
of.
(c) A set {w,...., w,} is a basis for V if, and only if, the set {w1,..., w*} is
a basis for V.
(d) If W is a subspace of V and W* consists of all elements w* for which the
corresponding vector w is in WI then W* is a subspace of V.
9-11 LINEAR MAPPINGS
If a function f maps a set X into a set Y, then we call X the domain of
the function, or mapping, f. The set of all values of f forms the range of the
mapping. The range of f is a subset of Y it may coincide with Y. We shall
call Y itself the target space of the mapping (Figure 9-15).
In the calculus we studied real functions f(x), defined on an interval; here
the domain is the interval (which we can consider as a subset of V1), the target space is V1, and the range is a subset of V1. We also considered vector
