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In Exercises 1–16 a set of objects is given, together with operations of addition and scalar multiplication. Determine which
sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold.
The set of all triples of real numbers (x, y, z) with the operations
1.
The set of all triples of real numbers (x, y, z) with the operations
2.
The set of all pairs of real numbers (x, y) with the operations
3.
The set of all real numbers x with the standard operations of addition and multiplication.
4.
The set of all pairs of real numbers of the form with the standard operations on .
5.
The set of all pairs of real numbers of the form , where , with the standard operations on .
6.
The set of all n-tuples of real numbers of the form with the standard operations on .
7.
The set of all pairs of real numbers with the operations
8.
The set of all matrices of the form
9.
with the standard matrix addition and scalar multiplication.
The set of all matrices of the form
10.
with the standard matrix addition and scalar multiplication. , with the operations
The set of all real-valued functions f defined everywhere on the real line and such that
11. defined in Example 4.
The set of all matrices of the form
12.
with matrix addition and scalar multiplication. with the operations
The set of all pairs of real numbers of the form
13.
