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

11. a. The set of all affine combinations of points in a set \(S\) is called the affine hull of \(S\).

b. If \(\left\{ {{{\rm{b}}_{\rm{1}}}{\rm{,}}.......{{\rm{b}}_{\rm{2}}}} \right\}\) is a linearly independent subset of \({\mathbb{R}^n}\) and if \({\bf{p}}\) is a linear combination of \({{\rm{b}}_{\rm{1}}}.......{{\rm{b}}_{\rm{k}}}\), then \({\rm{p}}\) is an affine combination of \({{\rm{b}}_{\rm{1}}}.......{{\rm{b}}_{\rm{k}}}\).

c. The affine hull of two distinct points is called a line.

d. A flat is a subspace.

e. A plane in \({\mathbb{R}^3}\) is a hyper plane.

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 True.

According to theorem 4,a point \(y\) in\({\mathbb{R}^n}\) is an affine combination of \({{\bf{v}}_1},.......,{{\bf{v}}_p}\)in \({\mathbb{R}^n}\) if \(\overline y = {c_1}{\overline {\bf{v}} _1} + ...... + {c_p}{\overline {\bf{v}} _p}\) such that \({c_1} + ...... + {c_p} = 1\).

Thus, the sum of weights in the linear combination should be 1.

So, statement in (b) is False.

The affine hull of \(\left\{ {{{\bf{v}}_1},\,{{\bf{v}}_2}} \right\}\)is the set \(y = \left( {1 - t} \right){{\bf{v}}_1} + t{{\bf{v}}_2}\). This equation represents a line.

So, the statement (c) is True.

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

So, statement (d) is False.

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

So, the statement (e) is True.