Page 119 - A Course in Linear Algebra with Applications

4.2: Vector Spaces and Subspaces 103

Exercises 4.2
1. Which of the following are vector spaces? The operations
of addition and scalar multiplication are the natural ones:
(a) the set of all 2 x 2 real matrices with determinant
equal to zero;
(b) the set of all solutions X of a linear system AX = B
where B ^ 0;
(c) the set of all functions y = y(x) that are solutions of
the homogeneous linear differential equation

a n(x)y {n) + a n^{x)y^- l) + ••• + ai{x)y' + a 0(x)y = 0.

2. In the following examples say whether S is a subspace of
the vector space V :
(a) V = R 2 and S is the subset of all matrices of the form

1 where a is an arbitrary real number;

(b) V = C[0,1] and S is the set of all infinitely
differentiable functions in V.
(c) V = -P(R) and S is the set of all polynomials p
such that p(l) = 0.
3. Does the polynomial 1 — 2x + x 2 belong to the subspace of
P3(R) generated by the polynomials 1 + X » X X and 3 — 2a:?

4. Determine if the matrix I 1 is in the subspace of

M2(R) generated by the following matrices:

3 4 \ / 0 2 \ / 0 2
1 2J' 1-1/3 4J' I 6 1

5. Prove that the vector spaces M m)Tl (F) and P n(F) are
finitely generated where F is an arbitrary field.