Question

W := {(x_1; x_2; x_3) in V l (x_2)=(x_1)(x_3)}
Is W a subspace of R^3 (subspace test)

Answer 1

there are certain criteria for "subspace"

Answer 2

do you recall what they are?

Answer 3

I know show 0 vector is in set, closure under addition and closure under multiplication, i have shown the first two but for the last i am unsure on whether i have done it right so just wanted to check

Answer 4

V l (x_2)=(x_1)(x_3)}

that is hard to parse thru, can you clear it up any?

Answer 5

\[ W:= \{(x_{1}, x_{2}, x_{3}) \in V \text{such that} x_{2}=x_{1}x_{3}\} ) \]

Answer 6

sorry wasnt sure on how to put a space after text

Answer 7

that is alot easier to read :)

Answer 8

if you can prove that:

\[c_1\vec X+c_2\vec Y\in W\]that should satisfy the addition and multiplcation requirements in one fell swoop

Answer 9

oh ok so i have done it seperately, but under multiplication can it be said that for cu=c(u1,u2,u3)
then
cu=(cu1,cu2,cu3)

and if we let say v= cu that conditions are met? as cu2=cu1cu3 implies v2=v1v3

if that makes sense.

Answer 10

cu1 cu2 = c^2 u1 u2

right?

Answer 11

yh see thats where i was not sure as then surely that wouldnt work

Answer 12

im getting the indexes a little mixed up i see

Answer 13

sorry its just my latex typing isnt that quick

Answer 14

\[\vec u=c_1\vec x_1+c_2\vec x_2+c_3\vec x_3\]
\[\vec u=c_1\vec x_1+c_2\vec x_1\vec x_3+c_3\vec x_3\]
\[k\vec u=k(c_1\vec x_1)+k(c_2\vec x_1\vec x_3)+k(c_3\vec x_3)\]

Answer 15

now im getting lost lol ... oy vey

Answer 16

ok i think i got it it let me type it in latex so it is clear.

Answer 17

Suppose \(c \in R \) and \(\mathbf{v} \in V \)

\[c\mathbf{v} = c(v_{1}, v_{2}, v_{3}) =(cv_{1}, cv_{2}, cv_{3}) \]Now if we let \(\mathbf{u} \in V \) where \(\mathbf{u} = c\mathbf{v}\) which is valid as it is given that \( (x_{1}, x_{2}, x_{3}) \in V \) and \(V\) is the vector space \(R^3\) so multiplying by a scalar will still be in the reals. Thus, we have that
\[ cv_{2} =cv_{1}cv_{3} \] and as \(\mathbf{u} = c\mathbf{v}\) it can be shown that
\[ u_{2} =u_{1}u_{3} \]

Answer 18

i think that makes sense.

Answer 19

with my limited recollection, that does make sense to me as well. but wouldnt we want: u in W? or is the line definition of v2 = v1v3 sufficient for that ....

Answer 20

im not even sure what that definition would actually entail ... but for the sake of definitions :)

Answer 21

yh you are right. i want u in W not V

Answer 22

by defining u in W, u = cv, and W is of the defintion x2= x1x3; cv2 = c^2 v1v2 for all c seems to defy closure to me

Answer 23

..but knowing me, id want a 2nd opinion :) @.Sam. you any decent at these things?

Answer 24

yh thats the reason i came on here to ask in the first place the c^2 might mean it is not a subspace.

Answer 25

if we could visualize a x1x3 product ... we would have something to compare with

Answer 26

if its a "dot product"

3<1,1,1>
3<4,1,0>
----------
3(4+1+0) doesnt seem to make sense to me

Answer 27

cross product the only other thing i can thing of for a test

Answer 28

<1,1,1>x<4,1,0> = <-1,4,-3>

<3,3,3>x<12,3,0> = <-9,36,-27> = 3<-3,12,-9>

Answer 29

in that case it definately doesnt hold right

Answer 30

thats what im wondering about we can still get <-1,4,-3> back again, by factoring out another 3, and since 3^2 is a real number ... im just not sure if thats acceptable or not :/

Answer 31

<-1,4,3> , if we are defining the product of x1x3 as a cross, IS in W, and any real scalar of it is as well

Answer 32

hmm i guess i will just have to email my lecturer see what he says

Answer 33

well, good luck with it :) i think ive stared myself into the 50s bracket of an IQ test with this one ....

Answer 34

well thanks for the help anyway, much appreciated