Question

Find an orthonormal basis q1, q2 for the column space of matrix A, C(A). Use Gram-Schmidt; show all of your steps.

a) Perform Gram-Schmidt on matrix A
b) Use the inverse of matrix A
c) Apply row reduction on matrix A
d) Utilize the determinant of matrix A

Answer 1

To find an orthonormal basis for the column space of matrix A using Gram-Schmidt, start by normalizing the first column vector, project the second column onto the first, subtract this projection to get an orthogonal vector, and normalize it. This yields an orthonormal basis for C(A), which consists of vectors q1 and q2.

Step-by-step explanation:

To find an orthonormal basis q1, q2 for the column space of matrix A, C(A), we utilize the Gram-Schmidt process. This process involves taking vectors that span the column space of A and orthogonalizing them. Assume that A has columns a1 and a2 that span C(A). Here are the steps of Gram-Schmidt:

1. Orthogonalization: Start with the first vector a1 and set it as your first basis vector for C(A), q1.
   q1 = a1 / ||a1||, where ||a1|| is the norm of a1.
2. Project a2 onto q1 to find the component of a2 in the direction of q1. projq1(a2) = (a2 · q1) * q1, where · denotes the dot product.
3. Subtract this projection from a2 to get a vector orthogonal to q1. u2 = a2 - projq1(a2).
4. Normalize u2 to get the second orthonormal basis vector. q2 = u2 / ||u2||.

The vectors q1 and q2 form an orthonormal basis for C(A). We do not use the inverse of matrix A, apply row reduction, or utilize the determinant in the Gram-Schmidt process. Instead, we focus on orthogonalization and normalization using dot products and vector norms.