# Homework Help: Problems about Subspaces

1. Jun 5, 2012

### Jimmy84

1. The problem statement, all variables and given/known data

-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.

2. Relevant equations

3. 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.

2. Jun 5, 2012

### 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...

3. Jun 5, 2012

### Jimmy84

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?

4. Jun 5, 2012

### algebrat

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.

5. Jun 5, 2012

### vela

Staff Emeritus
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?