# Math Algebra

posted by .

Transform the basis {[1, 0, 1], [0, 1, 2], [2, 1, 0]} for R^3 into an orthonormal basis, using the Gram-Schmidt process.

• Linear Algebra -

Define proj(u, v) to be the projection of u onto v. proj(u, v) = v*(u dot v)/(v dot v)

||u|| = norm u

From the 3 given vectors, we want to form a basis such that each basis vector is orthogonal to every other and an unit vector.

Take e1 = ||[1,0,1]|| = (1/sqrt(2))[1,0,1]. The first basis vector is arbitrary.

e2 = ||[0,1,2] - proj([0,1,2], [1,0,1])||. We remove the non-orthogonal part to ensure e2 is orthogonal. Then take the norm to make it an unit vector.

e3 = ||[2,1,0] - proj([2,1,0], [0,1,2]) - proj([2,1,0], [1,0,1])||. As with e2, remove every part orthogonal to e1 or e2 to ensure orthogonality. Then take the norm to make it an unit vector.

## Similar Questions

1. ### math

Find an orthonormal basis for the subspace of R^3 consisting of all vectors(a, b, c) such that a+b+c = 0. The subspace is two-dimensional, so you can solve the problem by finding one vector that satisfies the equation and then by constructing …
2. ### math

A trigonmetric polynomial of order n is t(x) = c0 + c1 * cos x + c2 * cos 2x + ... + cn * cos nx + d1 * sin x + d2 * sin 2x + ... + dn * sin nx The output vector space of such a function has the vector basis: { 1, cos x, cos 2x, ..., …
3. ### math

Find the least squares approximation of x over the interval [0,1] by a polynomial of the form a + b*e^x --------------------------------------------------------- The polynomial produces an output space with two linearly independent …
4. ### accounting

With this being 450.00 from Dec 31, 2003 to Dec, 31 2006 what would the accrual basis be and the cash basis for the balance sheet insurance asset using and the insurance expense using accrual basis and cash basis
5. ### ACC 225

What is the difference between cash basis and accrual basis accounting?
6. ### Accounting

What is the difference between cash basis and accrual basis accounting?
7. ### Math

1. P5 is an innerproduct space with an inner product. We applied the Gram Schmidt process tot he basis {1,x,x^2,x^3,x^4} and obtained the following as the result {f1,f2,f3,f4,x^4+2}. a. What is the orthogonal complement of P3 in P5 …
8. ### linear algebra check

Use the Gram-Schmidt process to transform the basis [1 1 1] , [0 1 1] , [2 4 3] for the Euclidean space R3 into an orthonormal basis for R3. (Enter each vector in the form [x1, x2, ...]. Enter your answers as a comma-separated list.) …
9. ### linear algebra urgent

Use the Gram-Schmidt process to transform the basis 1 1 1 , 0 1 1 , 2 4 3 for the Euclidean space R3 into an orthonormal basis for R3.
10. ### Linear Algebra

Hi, I really need help with these True/False questions: (a) If three vectors in R^3 are orthonormal then they form a basis in R^3. (b) If Q is square orthogonal matrix such that Q^2018 = I then Q^2017 = Q^T. (c) If B is square orthogonal …

More Similar Questions