Consider the family of vectors B 1 1 0 1 0 1 0 1 1 R3 Determ

Consider the family of vectors B:= {(1 1 0), (1 0 1), (0 1 1)} R^3. Determine whether or not the collection B is linearly independent family in R^3. Is the family B a basis of the (real) vector space R^3 under the standard addition and multiplication by scalars? (Justify your answer!) Determine whether B is a collection of orthogonal vectors. If not, orthogonalize it. (Show your work and state the name of the procedure you use!) Orthonormalize the family obtained in part (c) above.

Solution

Let v₁ = (1,1,0)^T , v₂ = =(1,0,1)^T , and v₃ = ( 0,1,1)^T. Let A be the matrix with v₁, v₂,v₃ as its 1^st, 2^nd and 3^rd columns. We will reduce A to its RREF as undeR:

1. Add -1 times the 1st row to the 2nd row; 2. Multiply the 2nd row by -1

3.. Add -1 times the 2nd row to the 3rd row; 4. Multiply the 3rd row by ½

5. Add 1 times the 3rd row to the 2nd row ; 6. Add -1 times the 2nd row to the 1st row

Then the RREF of A is I₃. This implies that:

a. The collection of vectors in B is a linearly independent family in R³.

b. The collection of vectors in B forms a basis for R³ under the standard addition and scalar multiplication.

c. Since v₁.v₂=(1,1,0)^T.(1,0,1)^T = 1 0, hence B is not a collection of orthogonal vectors. Let u₁ = v₁ , u₂ = v₂ – Proj_u1(v₂) = v₂ – [(v₂.u₁)/(u₁.u₁)]u₁ = (1,0,1)^T- 1/2(1,1,0)^T = (1/2, -1/2, 1)^T and u₃= v₃- Proj_u1(v₃) - Proj_u2(v₃) = v₃-[(v₃.u₁)/(u₁.u₁)]u₁-[(v₃.u₂)/(u₂.u₂)]u₂= v₃-(1/2)u₁- (1/3)u₂= (0,1,1)^T–(1/2,1/2,0)^T– (1/6, -1/6, 1/3)^T= (-2/3, 2/3, 2/3)^T. Thus, an orthogonal basis for B is { (1,1,0)^T, (1/2, -1/2, 1)^T, (-2/3, 2/3, 2/3)^T} . The Gram- Schimdt process has been used for orthogonalization.

d. Let w₁ = u₁/||u₁|| = (1/2, 1/2,0)^T , w₂ = u₂/||u₂||= (1/6 , -1/6, 2/3)^T, w₃ = u₃/||u₃|| = (-1/3, 1/3, 1/3)^T. Then { w₁ ,w₂, w₃} is an orthonormal basis for B.