Find a basis from cartesian equations

[parisse]

Inside Giac/Xcas, syst2mat does the conversion from linear equations to matrix form, unfortunately it is not available on the Prime.
m:=syst2mat([x1+2x2+x3+2x4+x5=0,-x1-2x2-x4+2x5=0,x1+2x2+3x3+4x4+7x5=0],[x1,x2,x3,x4,x5]);
ker(m[0..2,0..4]);