Determine if follow subsets are subspaces of vector space V

[Apprentice123]

Make sure the following subsets S are vector subspaces of the vector space V

Please check my answers

1) S={(x,y)/y=-x} V=\(\displaystyle R^2\)
u=(x1,-x1)
v=(x2,-x2)
u+v = (x1+x2,-x1-x2) = (x1+x2,-(x1+x2))
u+v E S

\(\displaystyle \alpha .u = \ alpha (x1,-x1) = ( \alpha x1,- \alpha x2 )\)
\(\displaystyle \alpha . u\) E S

Therefore, S is a vector subspace of V


2) S={(x,y,z)/y=z^2} V=R^3
u=(z1^2,y1,z1)
v=(z2^2,y2,z2)
u+v=
....
(z1^2+z2^2,y1+y2,z1+z2)
u+v Not E S

\(\displaystyle \alpha . u =\)
....
\(\displaystyle ( \alphaz1^2, \alpha y1, \alpha z1)\)

Not E S

 

[mmm4444bot]

Re: Vector subspaces

Make sure the following subsets S are vector subspaces of the vector space V

Please check my answers

1) S={(x,y)/y=-x} V=\(\displaystyle R^2\)
u=(x1,-x1)
v=(x2,-x2)
u+v = (x1+x2,-x1-x2) = (x1+x2,-(x1+x2))
u+v E S

\(\displaystyle \alpha .u = \ alpha (x1,-x1) = ( \alpha x1,- \alpha x2 )\)
\(\displaystyle \alpha . u\) E S

Therefore, S is a vector subspace of V ? THIS IS CORRECT


2) S={(x,y,z)/y=z^2} V=R^3
u=(z1^2,y1,z1)
v=(z2^2,y2,z2)
u+v=
....
(z1^2+z2^2,y1+y2,z1+z2) ? THIS IS NOT Y=Z^2

u+v Not E S ? THIS IS CORRECT (EITHER WAY)

 

[Apprentice123]

Re: Vector subspaces

Thanks you. But why y=z^2 ?

 

[pka]

Re: Vector subspaces

\(\displaystyle \begin{gathered} y_1 = z_1 ^2 \,\& \,y_2 = z_2 ^2 \hfill \\ \left( {y_1 + y_2 } \right) \ne \left( {z_1 + z_2 } \right)^2 \hfill \\ \end{gathered}\)

 

[mmm4444bot]

Re: Vector subspaces

But why y=z^2 ?

Because that is what you typed in the exercise.

~ Mark

 

[Apprentice123]

Re: Vector subspaces

But why y=z^2 ?

Because that is what you typed in the exercise.

~ Mark

Thanks