W is a subspace of R5. From equations x+y+z+w+t=0,x−y+z−w+t=0 we find: t=w−z+y−x . Substitute it in the first equation and get: x+y+z+w+w−z+y−x=2y+2w=0 . Thus, we get: t=−z−x, w=−y. Thus, the space is 3 dimensional. x,y,z are independent coordinates and coordinates t and w can be expressed via x,y,z . To find the basis we take the standard basis from R3: x1=1,y1=0,z1=0, x2=0,y2=1,z1=0 and x3=0,y3=0,z3=1 and extend it to R5 by using equations for t and w. We receive 3 vectors: e1=(1,0,0,0,−1),e2=(0,1,0,−1,0) and e3=(0,0,1,0,−1). This is a basis of W.
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