Show that {v1,v2,v3} is an orthonormal basis in R3 with the Euclidean inner product

Question

Consider Vector:

 

v1= 1/sqrt(18)  [  4 ]  ,   v2= 1/3  [ 1 ] ,   v3= 1/sqrt(2)  [0],   w= [2]

                       [ -1 ]                    [ 2 ]                          [1]         [1]              

                       [  1 ]                    [-2 ]                          [1]         [-1]

 

or it can be written as v1= 1/sqrt(18) (4,-1,1),    v2= 1/3 (1,2,-2),   v3= 1/sqrt(2) (0,1,1),  w= (2,1,-1)

 

 

 1)  Show that {v1,v2,v3} is an orthonormal basis in R3 with the Euclidean inner product  

 

2) Find the coordinates of vector w in this basis      

 

 

 

Answer 1

Hi Rola,

 

To show that B={v₁,v₂,v₃} is an orthogonal set, simply compute the inner products (dot products) of each pair of vectors and show it is 0.

 

To show that B is orthonormal, additionally, compute the magnitude of each vector and show it is 1.

 

To show that B is a basis, put the vectors into a 3x3 matrix as columns and compute the determinant, which should be nonzero.

 

To find the coordinates of w in the basis B, write

 

w = <w,v₁>v₁ + <w,v₂>v₂ + <w,v₃>v₃

 

Hope this gives you a good start.

 

Regards,

Hassan