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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

# In Exercises 1-6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points.1.\left( {\begin{aligned}{{}}3\\{ - 3}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\6\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\0\end{aligned}} \right)

The set of points is affinely dependent, and the relation is $$2{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2} - 3{{\mathop{\rm v}\nolimits} _3} = 0$$.

See the step by step solution

## Step 1: Condition for affinely dependent

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 $${c_1},{c_2},...,{c_p}$$ not all zero, and the sum must be zero $${c_1} + {c_2} + ... + {c_p} = 0$$, and $${c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0$$.

## Step 2:Compute $${{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1}$$ and $${{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}$$

Let {{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}3\\{ - 3}\end{aligned}} \right),{{\mathop{\rm v}\nolimits} _2} = \left( {\begin{aligned}{{}}0\\6\end{aligned}} \right),{{\mathop{\rm v}\nolimits} _3} = \left( {\begin{aligned}{{}}2\\0\end{aligned}} \right).

Compute translated points $${{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1}$$ and $${{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}$$as shown below:

{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}{ - 3}\\9\end{aligned}} \right),

{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}{ - 1}\\3\end{aligned}} \right)

It is observed that two points are multiples of each other.

## Step 3: Determine whether the set of points is affinely dependent

Theorem 5 states that an indexed set $$S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}$$ in $${\mathbb{R}^n}$$, with $$p \ge 2$$, the following statement is equivalent. This means that either all the statements are true or all the statements are false.

1. The set $$S$$ is affinely dependent.
2. Each of the points in $$S$$ is an affine combination of the other points in $$S$$.
3. In $${\mathbb{R}^n}$$, the set $$\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p} - {{\mathop{\rm v}\nolimits} _1}} \right\}$$is linearly dependent.
4. The set $$\left\{ {{{\bar v}_1},...,{{\bar v}_p}} \right\}$$ of homogeneous forms in $${\mathbb{R}^{n + 1}}$$ is linearly dependent.

Since two points are multiples of each other hence form a linearly dependent set.

Therefore, all statements in theorem 5 are true and thus $$S$$ are affinely dependent.

\begin{aligned}{}{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = 3\left( {{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}} \right)\\{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = 3{{\mathop{\rm v}\nolimits} _3} - 3{{\mathop{\rm v}\nolimits} _1}\\2{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2} - 3{{\mathop{\rm v}\nolimits} _3} = 0\end{aligned}

Thus, the set of points is affinely dependent and $$2{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2} - 3{{\mathop{\rm v}\nolimits} _3} = 0$$.