Question

Show that the set X with the given operations fails to be a vector space by identifying all axioms that hold and fail to hold:

The set X=R^3 with vector addition + defined by:
(a,b,c)+(x,y,z)=(1,y,c+z)
and scalar multiplication . defined by: k . (a,b,c)=(ka,kb,kc)

Answer 1

no zero vector for addition

Answer 2

there should be a vector, \(\vec{0}\), such that

\[<a,b,c>+\vec{0}=<a,b,c>\]

Answer 3

Correct me if I'm wrong Zarkon, but we also have no \(-\vec u\) such that \(\vec u+(-\vec u)=\vec0\) , right?
also we don't have \((c+k)\vec u=c\vec u+k\vec u\) because of the definition of addition here
so this is not a vector space for quite a few reasons

Answer 4

Correct...I just gave one reason why it was not a vector space.

Answer 5

thank you very much

Answer 6

Consider the vector space M22 of all 2*2 matrices.

2.1 Show that B ={A1,A2,A3,A4} is a basis for M22 where:
A1[3 6]; A2=[0 -1]; A3= [0 -8]; and A4= [1 0]
[3 -6] [-1 0] [-12 -4] [-1 2]

2.2 Write A=[6 2]
[5 3]

2.3 Prove that the subset

D={A element of M22: A^T+A=0}
forms a subspace of M22