Question

Let W be the subspace of R^2 spanned by the vector (1, 2). Using the standard inner product, let E be the orthogonal projection of R^2 onto W. Find an orthonormal basis in which E is represented by the matrix:

a. True
b. False

Answer 1

To find an orthonormal basis in which the orthogonal projection E is represented by a given matrix, we need to find the eigenvectors of the matrix representing E. The eigenvectors corresponding to non-zero eigenvalues are the vectors in the orthonormal basis.

Step-by-step explanation:

To find an orthonormal basis in which the orthogonal projection E is represented by the given matrix, we first need to find the eigenvectors of the matrix. Let A be the given matrix representing E in the standard basis. Since E is the orthogonal projection onto W, the matrix A is equal to the projection matrix onto W.

Next, we find the eigenvectors of A by solving the equation (A - λI)v = 0, where λ represents the eigenvalues of A and I is the identity matrix. The eigenvectors corresponding to non-zero eigenvalues are the vectors in the orthonormal basis.

In this case, the given matrix A is the projection matrix onto W, and since W is spanned by the vector (1, 2), A is equal to (1, 2)(1, 2)^T / ((1, 2)^T(1, 2)). By calculating the matrix A, we find its eigenvectors to obtain the orthonormal basis.