Chapter 1: Chapter 1

Problem 13

Give an example of a pair of non-collinear vectors in $R^{2}$. Then, show that the point $\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=(8,7)$ can be expressed as a linear combination of the non-collinear vectors.

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A pair of non-collinear vectors in $R^{2}$ could be $ \vec{a} = (1,0) $ and $ \vec{b} = (0,1) $. The point (8,7) can be expressed as a linear combination of these non-collinear vectors by finding scalars 's' and 't' such that $ s\vec{a} + t\vec{b} = (8,7) $. In this case, we have $s = 8$ and $t = 7$, so the linear combination is: (8, 7) = 8(1,0) + 7(0,1).

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TABLE OF CONTENTS

Step 1: Find a pair of non-collinear vectors in R^2.

Choose arbitrary vectors that are not scalar multiples of each other. One such possible pair could be: $ \vec{a} = (1,0) $ and $ \vec{b} = (0,1) $. Since their directions are different, these two vectors are non-collinear.

Step 2: Express the point (8,7) as a linear combination of the non-collinear vectors

Now we want to find scalars 's' and 't' such that: $ s\vec{a} + t\vec{b} = (8,7) $ Using the vectors defined in step 1, the equation becomes: $$s(1, 0) + t(0, 1) = (8, 7)$$ This simplifies to two separate equations: $$ s = 8 $$ $$ t = 7 $$ Since we found values for 's' and 't' that satisfy the equation, we can express (8,7) as a linear combination of the non-collinear vectors $\vec{a}$ and $\vec{b}$: $$ (8, 7) = 8(1,0) +7(0,1) $$

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Chapter 1: Chapter 1

Give an example of a pair of non-collinear vectors in \(R^{2}\). Then, show that the point \(\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=(8,7)\) can be expressed as a linear combination of the non-collinear vectors.

Short Answer

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Step 1: Find a pair of non-collinear vectors in R^2.

Step 2: Express the point (8,7) as a linear combination of the non-collinear vectors

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More chapters from the book ‘Linear Algebra’

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

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