How Do You Find the Basis and Equation for Specific Subspaces in R^3?

[Jimmy84]

Homework Statement

-Problem number 1

Given the set {u ,v} , where u=(1,2,1) and v=(0,-1,3) in R^3 find an equation for the space generated by this set.

-Problem number 2

The subspace S is defined as S= {(x,y,z) : x + 2y - z =0}
find a set B={u,v} in R^3 such that each vector in S is a linear combination of vectors in B.

Homework Equations

The Attempt at a Solution

I have no idea how to solve problem number 2.

I don't know how to find an equation for the space in problem one

I started the problem like this (a,b,c) =(1,2,1)x + (0,-1,3)y

and then I found x and y in terms of a and b. but I don't have an idea what is meant to find an equation for the space.

I would appreciate some help, thanks a lot.

 

[algebrat]

1. How about su+tv for real s and t?
2. Find two vectors in the plane. To do this, try z=0, et cetera...

 

[Jimmy84]

1. How about su+tv for real s and t?
2. Find two vectors in the plane. To do this, try z=0, et cetera...

the answer for the first problem is 7x -3y -z = 0 but I don't know how to solve for that

what do you mean s and t?

 

[algebrat]

the answer for the first problem is 7x -3y -z = 0 but I don't know how to solve for that

what do you mean s and t?

1. s and t would give the parametrized version of the plane. To get the implicit version they give, try the cross product, which gives the normal.

 

[vela]

-Problem number 2

The subspace S is defined as S= {(x,y,z) : x + 2y - z =0}
find a set B={u,v} in R^3 such that each vector in S is a linear combination of vectors in B.

If you have three variables and only one equation, you can solve for one variable in terms of the others. For example, if you had x-2y-3z=0, you could solve for x and get x=2y+3z. Now let y=s and z=t, where s and t are your free parameters, so you have
\begin{align*}
x &= 2s + 3t \\
y &= s \\
z &= t
\end{align*} Can you see how to write those three equations as one vector equation?