Linear algebra problem: find a unit vector in R ^ 4 so that it is orthogonal to the following three vectors: A1 (1,1, - 1,1), A2 (1, - 1, - 1,1), A3 (2,1,1,3) Request solution process, please help me | Math enthusiast | Mathematical art

Linear algebra problem: find a unit vector in R ^ 4 so that it is orthogonal to the following three vectors: A1 (1,1, - 1,1), A2 (1, - 1, - 1,1), A3 (2,1,1,3) Request solution process, please help me

Let x = (a, B, C, d) be orthogonal to A1, A2, A3, then a + B-C + D = 0, (1) a-b-c + D = 0, (2) 2A + B + C + 3D = 0, (3) (1) - (2) get b = 0, (2) + (3) get 3A + 4D = 0, take a = 4, then d = - 3, substitute (1) to get C = 1, so x = (4,0,1, - 3), calculate | X

It is known that A1, A2, A3 are linearly independent, B1 = a1 + A2, B2 = A2-A3, B3 = a1 + 2A3. It is proved that vector group B1, B2, B3 are linearly independent

(b1,b2,b3)=(a1+a2,a2-a3,a1+2a3) = (a1,a2,a3)K
K=
1 0 1
1 1 0
0 -1 2
Because | K | = 2-1 = 1 ≠ 0
So K is reversible
So r (B1, B2, B3) = R (A1, A2, A3) = 3
Therefore, B1, B2 and B3 are linearly independent

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