Help with vector spaces axioms

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  • #1

Homework Statement


for the 2x2 matrix [a 12;12 b] is it a vector space


Homework Equations


1. If u and v are objects in V, then u+v is in V
2. u+v = v + u
3. u+(v+w) = (u+v)+w
4. There is an object 0 in V, called a zero vector for V, such that 0+u = u+0 = u for all u in V
5. For each u in V, there is an object -u in V, called a negative of u, such that u+(-u) = (-u)+u = 0
6. If k is any scalar and u is any object in V, then ku is in V
7. k(u+v) = ku+kv
8. (k+m)u = ku+mu
9. k(mu) = (km)(u)
10. 1u = u


The Attempt at a Solution


axioms 1,4,5,6 fail to hold.

axiom 1:
for any vectors u = [u1,12; 12, u2], v = [v1,12 ; 12 v2]in the space, u + v must also be in the space
this is obviously false, since
u + v = [u1+v1,12 + 12; 12+12, u2 + v2]
= [u1+v1,24; 24, u2 + v2]
which is not in the space

axiom 4:
let u = [u1,12; 12, u2], we need to find a 0 vector such that
u + 0 = 0 + u = u
but 0 must have the form [z1,12; 12, z2] to be in the space
and so u + 0 = [u1,12; 12, u2] + [z1,12; 12, z2]
= [u1+z1,12+12 ; 12+12, u2+z2]
= [u1+z1, 24 ; 24, u2+z2]
which is not equal to vector u, regardless of z1 and z2 values.

axiom 5:
can't have an additive inverse, if there's no 0 vector

axiom 6:
let u = [u1,12; 12, u2]
according to this axiom, for any scalar k, k*u is also in the space
but this is false for any k =/= 1:
k*u = k*[u1,12; 12, u2]
=[k*u1,k*12; k*12, k*u2]
which is not in space unless k = 1

however I am not sure if I did this correctly and need confirmation.
 
Last edited:

Answers and Replies

  • #2
133
1
You did them all correctly. Note that you only need to find one property that doesn't work to conclude that V isn't a vector space. So you could have just done the first one and be done with it.
 
  • #3
thanks for the confirmation :)
 

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