1. ## find basis

Find a basis for each of the subspaces.

1. The subspace R^5 given by the solution space of the homogeneous system system

2x_1+2x_2-x_3+x_5=0
-x_1-x_2+2x_3-3x_4+x_5=0
x_1+x_2-2x_3-x_5=0
x_3+x_4+x_5=0.

2. The subspace of R^6 given by
W={(u,v,w,x,y,z) in R^6: u+w=v-x, v=y+z}

2. Hello,

$2x_1+2x_2-x_3+x_5=0$ (1)

$-x_1-x_2+2x_3-3x_4+x_5=0$ (2)

$x_1+x_2-2x_3-x_5=0$ (3)

$x_3+x_4+x_5=0.$ (4)

(2)+(3) :

$-3x_4=0 \rightarrow x_4=0$

So the system is resumed to :

$2x_1+2x_2-x_3+x_5=0$ (1)

$x_1+x_2-2x_3-x_5=0$ (3)

$x_3+x_5=0.$ (4')

$(4') \rightarrow x_3=-x_5$

$2x_1+2x_2+2x_5=0$ (1')

$x_1+x_2+x_5=0$

Now, keep the value for x_4=0.

Then replace x_5 by 0 for example and for 1. Then you will have x_3 thanks to (4')
And for x_1+x_2 do the same !