Q33E Among all the vectors in Rnwhose... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q33E (page 217)

Among all the vectors in $R^{n}$ whose components add up to 1, find the vector of minimal length. In the case $n = 2$ , explain your solution geometrically.

Short Answer

Expert verified

The vector with the minimal length is $\overset{r}{V} [\begin{matrix} 1 / n \\ 1 / n \\ . \\ . \\ 1 / n \end{matrix}]$

Geometrical representation is,

Step by step solution

Use Cauchy-Schwarz inequality

We can find such a vector of minimal length by using Cauchy- Schwarz inequality $| \overset{鈯�}{v} . \overset{鈯�}{w} | 鈮� || \overset{鈯�}{v} || . || \overset{鈯�}{w} ||$ .

Take,

$\overset{r}{V} [\begin{matrix} 1 / n \\ 1 / n \\ . \\ . \\ 1 / n \end{matrix}]$

By the above formula observe that $\overset{鈯�}{v .} \overset{鈯�}{w} = v_{1} + v_{2} + . . . + v_{n} = 1$ .

$鈬� 1 鈮� ||\overset{r}{v}|| \sqrt{n} 鈬� ||\overset{r}{v}|| 鈮� \frac{1}{\sqrt{n}}$

Also note that equality holds only and only if $\overset{鈯�}{u}$ and $\overset{鈯�}{w}$ are linearly dependent.

Thus,

$\overset{r}{v} [\begin{matrix} 位 \\ 位 \\ . \\ . \\ 位 \end{matrix}],$

$v_{1} + v_{2} + . . . + v_{n} = 1 鈬� n 位鈬� 位 = \frac{1}{n}$

consider the diagram

For $n = 2$ , $V_{1} + V_{2} = 1$ .

Now, draw the graph for equation $V_{1} + V_{2} = 1$ .

From the above figure it is clear that the vector of minimal length will be the vector which is perpendicular to the line $V_{1} + V_{2} = 1$ .

Hence, the vector with the minimal length will be $\overset{r}{V} [\begin{matrix} 1 / n \\ 1 / n \\ . \\ . \\ 1 / n \end{matrix}]$ .

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91影视

Among all the vectors in $R^{n}$ whose components add up to 1, find the vector of minimal length. In the case $n = 2$ , explain your solution geometrically.

Short Answer

Step by step solution

Use Cauchy-Schwarz inequality

consider the diagram

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91影视

Among all the vectors in Rnwhose components add up to 1, find the vector of minimal length. In the casen=2 , explain your solution geometrically.

Short Answer

Step by step solution

Use Cauchy-Schwarz inequality

consider the diagram

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Statistics

Pure Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Among all the vectors in $R^{n}$ whose components add up to 1, find the vector of minimal length. In the case $n = 2$ , explain your solution geometrically.