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Q11E

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Found in: Page 437

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Explain why any set of five or more points in $${\mathbb{R}^3}$$ must be affinely dependent.

Any set of five or more points in $${\mathbb{R}^3}$$ must be affinely dependent.

See the step by step solution

## Step 1: State the condition for affinely dependence

The set is said to be affinely dependent if the set $$\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_p}} \right\}$$ in the dimension$${\mathbb{R}^n}$$ exists such that for non-zero scalars$${c_1},{c_2},...,{c_p}$$, the sum of scalars is zero i.e.$${c_1} + {c_2} + ... + {c_p} = 0$$, and $${c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0$$.

## Step 2: Show affinely dependence

Consider the set of five points $$\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3},{{\bf{v}}_4},{{\bf{v}}_5}} \right\}$$.

Let $$\left\{ {{{\bf{v}}_2} - {{\bf{v}}_1},{{\bf{v}}_3} - {{\bf{v}}_1},{{\bf{v}}_4} - {{\bf{v}}_1},{{\bf{v}}_5} - {{\bf{v}}_1}} \right\}$$ be the set of vectors in $${\mathbb{R}^3}$$.This new set of four points must be linearly dependent when a set of vectors is translated by eliminating the first point or any other point.

From $${\mathbb{R}^3}$$, the number of entries is 3. From the above-considered set of points, the number of vectors is 4. That is, $$n = 3$$, and $$p = 4$$.

Here, the set contains more vectors or points than the number of entries. So,$$p > n$$and the original set of five points $$\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3},{{\bf{v}}_4},{{\bf{v}}_5}} \right\}$$ is affinely dependent.

Therefore, any set of five or more points in $${\mathbb{R}^3}$$ must be affinely dependent.