Let v 5 3 6 8 Find a basis of the subspace of R4 consistin

Let v = [5 3 - 6 8]. Find a basis of the subspace of R^4 consisting of all vectors perpendicular to v.

Solution

let a vector be ( x , y , z , w )

then we can write

5x + 3y - 6z + 8w = 0

w = (5x + 3y - 6z )/ 8

we can write

( x , y , z , (5x + 3y - 6z )/ 8 )

this is teh form of all vectors perpendicular to the given vector

= x ( 1,0,0 , 5/8 ) + y ( 0,1,0 , 3/8 ) , z ( 0,0,1 , -6/8 )

hence basis is { ( 1,0,0 , 5/8 ) , ( 0,1,0 , 3/8 ), ( 0,0,1 , -3/4 ) }