Question

Question: 12. In Exercises 11 and 12, mark each statement True or False. Justify each answer.

a. If \(S = \left\{ {\bf{x}} \right\}\), then \({\rm{aff}}\,S\) is the empty set.

b. A set is affine if and only if it contains its affine hull.

c. A flat of dimension 1 is called a line.

d. A flat of dimension 2 is called a hyper plane.

e. A flat through the origin is a subspace.

Short answer

1. The given statement is False.
2. The given statement is True.
3. The given statement is True.
4. The given statement is False.
5. The given statement is True.

Step-by-step solution

Use the definition of affine combination

According to the definition of affine combination, the set of all affine combinations of points in a set S is called the affine hull (or affine span) of \(S\), denoted by \({\rm{aff }}S\).

So, the statement (a) is False.

Use theorem 2

According to theorem 2,a set \(S\) is affine if and only if every affine combination of points of \(S\) lies in \(S\).

That is, \(S\) is affine if and only if \(S = {\rm{aff}}S\).

So, statement (b) is True.

Use the concept of proper subspaces in \({\mathbb{R}^3}\)

The proper flats in \({\mathbb{R}^3}\) are points (zero-dimensional), lines (one-dimensional), and planes, but not hyperplanes (two-dimensional), which may or may not pass through the origin.

So, statement in (c) is True, whereas (d) is False.

Use the concept of flat

A flat through the origin is a subspace only, which is translated by the 0 vectors.

So, statement in (e) is True.