Question

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)

Answer 1

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

Answer 2

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

Answer 3

so we are finding what order space this is?

Answer 4

mm.. what u mean with order space?

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

2)scalar multiplication***

Answer 9

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

Answer 10

are you learning it independently or with class?

Answer 11

class

Answer 12

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

Answer 13

so in this case im trying to verify number 8

Answer 14

distributive property

Answer 15

kind of

Answer 16

do you watch MIT lectures

Answer 17

mmm
nop

Answer 18

are u learning independenly

Answer 19

yes

Answer 20

ohh

Answer 21

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

Answer 22

have you done differential equation yet?

Answer 23

yes in calculus..

Answer 24

im learning it now

Answer 25

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

Answer 26

:).. 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

Answer 27

no, I just finished the class this semseter

Answer 28

so u r good with double and triples integrals too?

Answer 29

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