Question

Find a homogeneous system whose solution space is spanned by

Answer 1

Answer:

Let

v1=(1,−2,0,3,−1),v2=(2,−3,2,5,−3),v3=(1,−2,1,2,−2).

and L=span(v1,v2,v3). Let's find L⊥={x∣∀v∈L:(v,x)=0}:

L⊥:⎧⎩⎨x1−2x2+3x4−x52x1−3x2+2x3+5x4−3x5x1−2x2+x3+2x4−2x5=0,=0,=0.

Solve this system:

x1x2x3x4x5=t1,=t1+4t2,=−t1−4t2,=t2,=−t1−5t2,

or

x=⎛⎝⎜⎜⎜⎜⎜⎜11−10−1⎞⎠⎟⎟⎟⎟⎟⎟t1+⎛⎝⎜⎜⎜⎜⎜⎜04−41−5⎞⎠⎟⎟⎟⎟⎟⎟t2=u1t1+u2t2.

But L=(L⊥)⊥. Do the same trick and

L:{x1+x2−x3−x54x2−4x3+x4−5x5=0=0

is needed system.