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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)
I am learning linear algebra as well, can you tell me what topic this comes from ?
determining is v is a vector space... so we have to verify the ten axioms
so we are finding what order space this is?
mm.. what u mean with order space?
line plane is 2nd order space, line is 1st order space
ohhh,,, well my example is already i 3 space
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
2)scalar multiplication***
I know about first two , but never heard of this '10 axioms' stuff
are you learning it independently or with class?
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
so in this case im trying to verify number 8
distributive property
do you watch MIT lectures
are u learning independenly
I tend to do better in class , if I already learned it beforehand
have you done differential equation yet?
let me know if you need any help with differential equation; I am really good at it
:).. 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
no, I just finished the class this semseter
so u r good with double and triples integrals too?
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