Page 347 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The set of polynomials of the form with the operations
14.

The set of all positive real numbers with the operations
15.

The set of all pairs of real numbers with the operations
16.

Show that the following sets with the given operations fail to be vector spaces by identifying all axioms that fail to hold.
17.

(a) The set of all triples of real numbers with the standard vector addition but with scalar multiplication defined by
.

(b) The set of all triples of real numbers with addition defined by and
standard scalar multiplication.

(c) The set of all invertible matrices with the standard matrix addition and scalar multiplication.

18. Show that the set of all matrices of the form with addition defined by
is a vector space. What is the zero vector in this space?
and scalar multiplication defined by

19.
(a) Show that the set of all points in lying on a line is a vector space, with respect to the standard operations of
vector addition and scalar multiplication, exactly when the line passes through the origin.

(b) Show that the set of all points in lying on a plane is a vector space, with respect to the standard operations of
vector addition and scalar multiplication, exactly when the plane passes through the origin.

Consider the set of all invertible matrices with vector addition defined to be matrix multiplication and the standard
20. scalar multiplication. Is this a vector space?

Show that the first nine vector space axioms are satisfied if has the addition and scalar multiplication operations
21. defined in Example 5.

Prove that a line passing through the origin in is a vector space under the standard operations on .
22.