A First Course in Linear Algebra, 2017a

9.2. Spanning Sets 465
Show that, with this definition of the vector space operations that P 2 is a vector space. Now let V denote
those polynomials a + bx + cx 2 such that a + b + c = 0. Is V a subspace of P 2 ? Explain.
Exercise 9.1.17 Let M,N be subspaces of a vector space V and consider M + N defined as the set of all
m + nwherem∈ M and n ∈ N. Show that M + N is a subspace of V.
Exercise 9.1.18 Let M,N be subspaces of a vector space V . Then M ∩ N consists of all vectors which are
in both M and N. Show that M ∩ N is a subspace of V.
Exercise 9.1.19 Let M,N be subspaces of a vector space R 2 . Then N ∪M consists of all vectors which are
in either M or N. Show that N ∪ M is not necessarily a subspace of R 2 by giving an example where N ∪ M
fails to be a subspace.
Exercise 9.1.20 Let X consist of the real valued functions which are defined on an interval [a,b]. For
f ,g ∈ X, f + g is the name of the function which satisfies ( f + g)(x)= f (x)+g(x). For s a real number,
(sf)(x)=s( f (x)). Show this is a vector space.
Exercise 9.1.21 Consider functions defined on {1,2,···,n} having values in R. Explain how, if V is the
set of all such functions, V can be considered as R n .
Exercise 9.1.22 Let the vectors be polynomials of degree no more than 3. Show that with the usual
definitions of scalar multiplication and addition wherein, for p(x) a polynomial, (ap)(x)=ap(x) and for
p,q polynomials (p + q)(x)=p(x)+q(x), this is a vector space.
9.2 Spanning Sets
Outcomes
A. Determine if a vector is within a given span.
In this section we will examine the concept of spanning introduced earlier in terms of R n . Here, we
will discuss these concepts in terms of abstract vector spaces.
Consider the following definition.
Definition 9.11: Subset
Let X and Y be two sets. If all elements of X are also elements of Y then we say that X is a subset
of Y and we write
X ⊆ Y
In particular, we often speak of subsets of a vector space, such as X ⊆ V . By this we mean that every
element in the set X is contained in the vector space V .