⎝⎛1233−3−2−312⎠⎞⎝⎛xyz⎠⎞=⎝⎛000⎠⎞
Performing row operations on the matrix, we have
⎝⎛1233−3−2−312⎠⎞→R31(−3)R21(−2)⎝⎛1003−9−11−3711⎠⎞
→R2(9−1)⎝⎛10031−11−39−711⎠⎞→R32(11)⎝⎛100310−39−7922⎠⎞
→R3(229)⎝⎛100310−39−71⎠⎞
x+3y−3z=0y−97z=0z=0
Therefore x=0,y=0, z=0. This is a trivial solution hence it is not a basis
Question B
⎝⎛1231341−10112⎠⎞⎝⎛xyzw⎠⎞=⎝⎛000⎠⎞
The matrix of the coefficient is
⎝⎛1231341−10112⎠⎞→R31(−3)R21(−2)⎝⎛1001111−3−31−1−1⎠⎞→R32(−1)⎝⎛1001101−301−10⎠⎞
x+y+z+w=0y−3z−w=0
Let z=a,w=b where a,b∈R
So, we have
y−3a−b=0y=3a+bx=−y−z−wx=−(3a+b)−a−bx=−4a−2b∴x=−4a−2b,y=3a+b,z=a,w=b(x,y,z,w)=a(−4,3,1,0)+b(−2,1,0,1)
So a basis for the system is {(-4,3,1,0),(-2,1,0,1)}
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