Question

Find a basis for the following subspace of R^4. All vectors of the form (a+c, a-b, b+c, -a+b).

Answer 1

i would set it up so that they all have an a,b,c component

Answer 2

So far I have
\[\left(\begin{matrix}a+c \\ a-b \\ b+c \\ -a+b\end{matrix}\right)=a \left(\begin{matrix}1 \\ 1 \\0 \\-1\end{matrix}\right)+b \left(\begin{matrix}0 \\ -1 \\1 \\1\end{matrix}\right)+c \left(\begin{matrix}1 \\ 0 \\1 \\0\end{matrix}\right)\]

Answer 3

1a + 0b + 1c
1a - 1b + 0c
0a + 1b + 1c
-1a + 1b + 0c

then strip the a,b,c parts and row reduce

1 0 1
1 -1 0
0 1 1
-1 1 0

Answer 4

a basis, if i recall, is a most efficient span; so we would have to determine the pivot columns

Answer 5

1 0 1
1 -1 0
0 1 1
-1 1 0


1 0 1
-1 1 0
0 1 1
0 1 1

1 0 1
0 1 1
0 1 1
0 1 1

1 0 1
0 1 1
0 0 0
0 0 0

looks like we have a free variable; and 2 pivot columns

Answer 6

basis = [1 1 0 -1], [0,-1,1,1]

might be it

Answer 7

opps, got comma happy :) those are spose to be column vectors

basis = [1 1 0 -1], [0 -1 1 1]

might be it

Answer 8

@amistre64 agreed