# Squaring a vector and subspace axioms

If a problem I'm doing asks to find

V2 where V is a vector

is it simply the dot product of the vector, or the cross product?

The question: Which of the following sets of vectors v = {v1,...,vn} in Rn are subspaces of Rn (n>=3)

iii) All v such that V2=V12

He proved it by saying it's not closed under addition (axiom of a subspace)

By (1,1,0) + (-1,1,0) = (0,2,0)

And that concludes his proof, but I'm not seeing what he's proved there at all.

Personally I would have done let a = (a1,...,an) and b = (b1,...,bn)

then a+b = (a1+b1,....,an+bn)

and so (a+b)2=(a1+b1,....,an+bn)2=(a1+b1,....,an+bn).(a1+b1,....,an+bn)

= a12+2a1b1+b12+.. (dot product)

Which is a scalar field not a vector field, so it's not closed under addition.

Am I wrong?

Last edited:

Dick
Homework Helper
They aren't asking you to square the vector. The v's are the components of the vector (v1,v2,v3). The question is take the subset of these vectors such that v2=v1^2. Like (2,4,1) and the first two vectors in your example.

They aren't asking you to square the vector.
The v's are the components of the vector (v1,v2,v3). The question is take the subset of these vectors such that v2=v1^2. Like (2,4,1) and the first two vectors in your example.

I was assuming v was a matrix of vectors, and each v1 etc was a vector. hence '...following sets of vectors v = {v1,...,vn}'

I still don't understand the rest, especially his proof

Dick