8.
x1=3x2 and x3=7x4
⎝⎛3x2x27x4x4x5⎠⎞=x2⎝⎛31000⎠⎞+x4⎝⎛00710⎠⎞+x5⎝⎛00001⎠⎞
Basis of W is (3,1,0,0,0),(0,0,7,1,0),(0,0,0,0,1)
9.
x1+x2−x3=0.....(i)x1−x3=0......(ii)−2x1−x2+2x3=0....(iii)−2x1+2x3=0....(iv)
−2(ii)=(iv), therefore ignoring (iv)
⎝⎛11−210−1−1−12⎠⎞⎝⎛x1x2x3⎠⎞=⎝⎛000⎠⎞
⟹⎝⎛100010−100⎠⎞⎝⎛x1x2x3⎠⎞=⎝⎛000⎠⎞
x1=x3x2=0
⎝⎛x10x1⎠⎞=x1⎝⎛101⎠⎞
Basis is [1,0,1]
10.
⎝⎛12−300024−6−3−69⎠⎞
21R2→R2
3−1R3→R3
⎝⎛111000222−3−3−3⎠⎞
R2−R1→R2R3−R1→R3
⎝⎛100000200−300⎠⎞
Rank =1
Nullity =3
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