The give set is a basis for a subspace W Use the Gramschmidt

The give set is a basis for a subspace W. Use the Gram-schmidt process to produce an orthogonal basis for W x_1 = [0 1 -1 1], x_2 = [1 1 -1 -1], x_3 = [1 0 1 1] A. [0 1 -1 1],[3 2 -2 -4], [14 2 9 7] B.[0 1 -1 1], [1 0 0 -2], [6 0 1 3] C.[0 1 -1 1],[3 4 -4 -2], [18 4 19 13] D.[0 1 -1 1], [1 1 -1 -1], [1 0 1 1]

Solution

the first set is the required set since just try to find the dot product of each combination of vectors you can see that the dot product is zero

orthogoality means just dot product is zero

example

in the first set if you take the dot product of first vector a nd second vector we have

0*3+1*2+-1*-2+-1*4 = 0

similary if you take first and third vector we have dot product to be 0*14+1*2+-1*9+1*7 = 0

similarlly third vector and second vector also we have the dot product is zero

so its just enough to check rather going on for gram schmidt process