LINEAR ALGEBRA. verifying (k+m)(u)=ku+mu... where k,m are any integers and u is an element of V={(x,y,z) | z=3}. addition defined by (x1,y1,z1)+(x2,y2,z2) = (x1+x2,y1+y2,3) scalar multiplication defined by s(x,y,z)=(sx,sy,3)

- anonymous

- chestercat

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- anonymous

I am learning linear algebra as well, can you tell me what topic this comes from ?

- anonymous

determining is v is a vector space... so we have to verify the ten axioms

- anonymous

so we are finding what order space this is?

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- anonymous

mm.. what u mean with order space?

- anonymous

line plane is 2nd order space, line is 1st order space

- anonymous

ohhh,,, well my example is already i 3 space

- anonymous

if u already covered this section.. we defined vector space as a set of vectors on which are defined by two operations:
1) vector addition
2) vector multiplication
and must satisfy 10 axioms

- anonymous

2)scalar multiplication***

- anonymous

I know about first two , but never heard of this '10 axioms' stuff

- anonymous

are you learning it independently or with class?

- anonymous

class

- anonymous

1. u + v 2 V (V is closed under addition)
2. u + v = v + u (commutative property)
3. (u + v) + w = u + (v + w) (associative property)
4. There exists a vector in V , called the zero vector and denoted 0 such that
u + 0 = u (additive identity)
5. For every vector u in V , there exists a vector u also in V such that
u + (u) = 0 (additive inverse)
6. cu 2 V (V is closed under scalar multiplication)
7. c (u + v) = cu + cv
8. (c + d) u = cu + du
9. c (du) = (cd) u
10. 1u = u

- anonymous

so in this case im trying to verify number 8

- anonymous

distributive property

- anonymous

kind of

- anonymous

do you watch MIT lectures

- anonymous

mmm
nop

- anonymous

are u learning independenly

- anonymous

yes

- anonymous

ohh

- anonymous

I tend to do better in class , if I already learned it beforehand

- anonymous

have you done differential equation yet?

- anonymous

yes in calculus..

- anonymous

im learning it now

- anonymous

let me know if you need any help with differential equation; I am really good at it

- anonymous

:).. cool... i like derivative for some reasons... where im having a little trouble is in double and triple integrals... are u also learning that independnly

- anonymous

no, I just finished the class this semseter

- anonymous

so u r good with double and triples integrals too?

- mathmate

Given u(x,y,z),
(x1,y1,z1)+(x2,y2,z2) = (x1+x2,y1+y2,3)
s(x,y,z)=(sx,sy,3)
Assuming vector space is defined over field F such as R.
(k+m)(u)
=(k+m)(x,y,3) [definition of u]
=((k+m)x,(k+m)y,3) [definition of scalar multiplication]
=(kx+mx, ky+my,3) [distributivity defined over F]
=(kx,ky,3)+(mx,my,3) [definition of addition]
=k(x,y,3)+m(x,y,3) [definition of scalar multiplication]
=ku+mu [definition of u] QED

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