# Let V=R³ and W={(x,y,z)|x+y+z=0} be a subspace of V.Which of the following pairs of vectors are in the same coset of W in V? (i)(1,3,2)and(2,2,2).(ii)(1,1,1)and(3,3,3).

Feb 5, 2018

$\setminus$

 \mbox{i)} \ (1,3,2) \ \mbox{and} \ (2,2,2):

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus m b \otimes \left\{\mathrm{do} b e l o n g \to t h e s a m e \cos e t o f\right\} \setminus W .$

 \mbox{ii)} \ (1,1,1) \ \mbox{and} \ (3,3,3):

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus m b \otimes \left\{\mathrm{do} \neg b e l o n g \to t h e s a m e \cos e t o f\right\} \setminus W .$

#### Explanation:

$\setminus$

 \mbox{1) Note that, by the given on} \ \ W, \mbox{we may describe} \ \mbox{the elements of} \ \ W \ \mbox{as those vectors of} \ \ V \ \mbox{where the} \ \mbox{sum of the coordinates is} \ 0.

$\setminus$

 \mbox{2) Now recall that:}

$\setminus m b \otimes \left\{t w o \vec{\to} r s b e l o n g \to t h e s a m e \cos e t o f a n y \subset s p a c e\right\}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \iff$

$\setminus q \quad \setminus m b \otimes \left\{t h e i r \mathrm{di} f f e r e n c e b e l o n g s \to t h e \subset s p a c e i t s e l f\right\} .$

$\setminus$

 \mbox{3) Thus to determine membership in the same coset of} \ W, \ \mbox{it is necessary and sufficient to determine if the} \ \mbox{difference of those vectors belong to} \ W :

$\setminus q \quad \setminus \vec{{v}_{1}} , \setminus \setminus \vec{{v}_{2}} \setminus \setminus \in \setminus \setminus m b \otimes \left\{s a m e \cos e t o f\right\} \setminus W \setminus \quad \setminus \iff \setminus \quad \setminus \vec{{v}_{1}} - \setminus \vec{{v}_{2}} \setminus \setminus \in \setminus W .$

$\setminus$

$\setminus m b \otimes \left\{H e n c e , b y t h e \mathrm{de} s c r i p t i o n o f\right\} \setminus W \setminus \setminus m b \otimes \left\{\in \left(1\right) a b o v e , w e h a v e\right.$

$\setminus \vec{{v}_{1}} , \setminus \setminus \vec{{v}_{2}} \setminus \setminus \in \setminus \setminus m b \otimes \left\{s a m e \cos e t o f\right\} \setminus W \setminus \quad \setminus \iff \setminus \quad \setminus m b \otimes \left\{t h e \sum o f t h e c \infty r \mathrm{di} n a t e s o f\right\} \setminus \setminus \left(\setminus \vec{{v}_{1}} - \setminus \vec{{v}_{2}}\right) = 0.$

$\setminus$

$\setminus m b \otimes \left\{I t i s a m a \texttt{e} r o f t h i s s i m p \le c o m p u t a t i o n .\right\}$

$\setminus$

 4) \ \mbox{Proceeding with the two given pairs of vectors, and} \ \mbox{performing this computation on each pair, we find:

 \quad \mbox{i)} \ \ (1,3,2) - (2,2,2) = (-1,1,0), \ \mbox{and so}

$\setminus q \quad \setminus q \quad \setminus m b \otimes \left\{t h e \sum o f t h e c \infty r \mathrm{di} n a t e s o f\right\} \setminus \quad \left(- 1 , 1 , 0\right) = 0.$

$\setminus m b \otimes \left\{H e n c e\right. \setminus q \quad \setminus q \quad \setminus q \quad \left(1 , 3 , 2\right) \setminus \setminus m b \otimes \left\{\mathmr{and}\right\} \setminus \left(2 , 2 , 2\right)$
$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus m b \otimes \left\{b e l o n g \to t h e s a m e \cos e t o f\right\} \setminus W .$

$\setminus$

 \quad \mbox{ii)} \ \ (1,1,1) - (3,3,3) = (2,2,2), \ \mbox{and so}

$\setminus q \quad \setminus q \quad \setminus m b \otimes \left\{t h e \sum o f t h e c \infty r \mathrm{di} n a t e s o f\right\} \setminus \quad \left(2 , 2 , 2\right) = 6 \setminus \ne 0.$

$\setminus m b \otimes \left\{H e n c e\right. \setminus q \quad \setminus q \quad \setminus q \quad \left(1 , 1 , 1\right) \setminus \setminus m b \otimes \left\{\mathmr{and}\right\} \setminus \left(3 , 3 , 3\right)$
$\setminus q \quad \setminus \quad \setminus \quad \setminus \setminus \setminus m b \otimes \left\{\mathrm{do} \neg b e l o n g \to t h e s a m e \cos e t o f\right\} \setminus W .$