Question

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.

Short answer

1. The given statement is True.
2. The given statement is False.
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 True.

Use theorem 4

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.

Use the affine concept hull

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.

Use the concept of flat

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

So, statement (d) is False.

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 (two-dimensional), which may or may not pass through the origin.

So, the statement (e) is True.