Linear Algebra Re: Subspaces ({(x1,x2)T|x1+x2=0})

[chucknorrisfish]

I need some help with the basics of figuring out subspaces. For Example:

Determine if {(x1,x2)T | x1 + x2 = 0 } forms a subspace of R2...

I'm not sure where to begin on how to test this out. Also, what the heck is the trivial solution?

Any help is appreciated, thanks!

 

[galactus]

Re: Linear Algebra Re: Subspaces

Every homogeneous system of linear equations is consistent, since all such systems have \(\displaystyle x_{1}=0, \;\ x_{2}=0,............, \;\ x_{n}=0\) as a solution. This solution is the trivial solution.

Everything else is non-trivial.

 

[merlin2007]

In order to determine whether the given set is a subspace, you have to check three conditions.
1) That the vector \(\displaystyle \[{\mathbf{0}}\]\) is an element of the set.
2) That the system is closed under scalar multiplication. If \(\displaystyle \[c \in \mathbb{R},{\mathbf{v}} \in V,c{\mathbf{v}} \in V\]\).
3) That the system is closed under addition. If \(\displaystyle \[{\mathbf{u}},{\mathbf{v}} \in V\]\), then \(\displaystyle \[{\mathbf{u}} + {\mathbf{v}} \in V\]\).

If you verify these three conditions, then your set is a subspace.