Question

Apply the Gram-Schmidt orthonormalization process to transform the given basis for Rn into an orthonormal basis. Use the vectors in the order in which they are given.

B = {(0, 1), (5, 7)}
u1 =

u2 =
ed

Answer 1

Explanation:

We can use the Gram-Schmidt orthonormalization process to transform the given basis B = {(0, 1), (5, 7)} for R^2 into an orthonormal basis.

First, we set u1 = (0, 1), the first vector in B. Then, we subtract the projection of the second vector in B onto u1 to get u2.

u1 = (0, 1)

u2 = (5, 7) - ((5, 7) dot (0, 1)) / ((0, 1) dot (0, 1)) * (0, 1)

= (5, 7) - 7 / 1 * (0, 1)

= (5, 0)

Next, we normalize u1 and u2 by dividing each vector by its length:

u1 = (0, 1) / ||(0, 1)|| = (0, 1)

u2 = (5, 0) / ||(5, 0)|| = (1, 0)

Therefore, an orthonormal basis for R^2 using the given basis B = {(0, 1), (5, 7)} is {u1, u2} = {(0, 1), (1, 0)}.