1. ## basis for subspaces

How do you find a basis for the given subspaces of $\displaystyle R^3$ and $\displaystyle R^4$.
a. All vectors of the form (a,b,c), where a=0
b. All vectors of the form (a+b, a-b, b+c, -a+b)
c. All vectors of the form (a,b,c), where a-b+5c=0

2. Originally Posted by antman
How do you find a basis for the given subspaces of $\displaystyle R^3$ and $\displaystyle R^4$.
a. All vectors of the form (a,b,c), where a=0
$\displaystyle (0,b,c) = (0,b,0) + (0,0,c) = b(0,1,0) + c(0,0,1)$
b. All vectors of the form (a+b, a-b, b+c, -a+b)
$\displaystyle (a,a,0,-a) + (b,-b,b,b) + (0,0,c,0) = a(1,1,0,-1) + b(1,-1,1,1) + c(0,0,1,0)$
c. All vectors of the form (a,b,c), where a-b+5c=0
$\displaystyle (a,b,c) = (a,a+5c,c) = a(1,1,0) + c(0,5,1)$.

3. I understand how you did this problem, but I am confused about your anser to part b:

Shouldn't this be (a,a,0,-a)+(0,-b,b,b)+(c,0,c,0)=a(1,1,0,-1)+b(0,-1,1,1)+c(1,0,1,0)
Set={(1,1,0,-1),(0,-1,1,1),(1,0,1,0)} for all vectors of the form (a+b, a-b, b+c, -a+b)? Thank you!

4. Originally Posted by antman
I understand how you did this problem, but I am confused about your anser to part b:

Shouldn't this be (a,a,0,-a)+(0,-b,b,b)+(c,0,c,0)=a(1,1,0,-1)+b(0,-1,1,1)+c(1,0,1,0)
Set={(1,1,0,-1),(0,-1,1,1),(1,0,1,0)} for all vectors of the form (a+b, a-b, b+c, -a+b)? Thank you!
Just add up the components together to see if they match up. Add up the first coordinate, second, third, and fourth. You should get $\displaystyle (a+b, a-b, b+c, -a+b)$ which is what I get.

5. I apologize. You are correct but I made a mistake on the problem. The vectors were suppoed to be the form of (a+c,a-b,b+c,-a+b). I don't know why I didn't notice my typo. Thank you.