Determine whether the following sets form subspaces of Rn
a.{( x1, x2)T | x1+x2 =0}
b. {( x1, x2)T | x1x2 = 0}

Question

Determine whether the following sets form subspaces of Rn 
a.{( x1, x2)T | x1+x2 =0}
b. {( x1, x2)T | x1x2 = 0}

jamesj · Answer

Draw these two subsets of R^2
a) is straight line
b) is two straight lines

This should suggest to you that a probably is but b has problems

anonymous · Answer

Thanks Jmaes but what do u mean b have a problem and what kind or equation can be used to find the subspace besides a graphical represntation. Dnt we need to prove a linear combination exist?

anonymous · Answer

sorry, i meant James

jamesj · Answer

The vector space is 2-dimenional Cartesian space, a plane, R^2.

Then think of x1 as the x value, x2 as the y value.

With that formulation, the first set is

x + y = 0.  

This is the straight line y = -x passing through the origin.  This looks like a sub-space.

But you need to prove this formally using the axioms of a subspace.

====

Now what is the second set in R^2.  At least one subspace axiom is violated; this is not a subspace.

anonymous · Answer

oh ok thanks i think i get the picture gna try these now c. {( x1, x2)T | x1 = 3x2 }
d. {( x1, x2)T | x12 = x22}

jamesj · Answer

Geometrically c is also a straight line through the origin.  d is not.