Consider the matrix given. Explain why it is not orthonormal, then use the columns with Gram-Schmidt to create an orthonormal basis for its column space.
-1
-1 -1 1
1 3 3
-1 -1 5
1 3 7
This is the given matrix
To determine if a matrix is orthonormal, we need to check two properties:
1. The columns (or rows) must be orthogonal to each other.
2. The length (or magnitude) of each column (or row) must be 1.
Given that you have only provided the value "-1," I will assume that you are referring to a square matrix with a single column of size n x 1.
Let's call this matrix A. Since A only has one column, it can be written as [a₁] with dimensions n x 1.
To check for orthonormality of A, we need to examine both properties:
1. The dot product of a column with itself must be 1: a₁·a₁ = 1.
However, since we only have the value "-1" and no other information, we cannot determine whether this condition is satisfied. Thus, it is not orthonormal.
To create an orthonormal basis for its column space using the Gram-Schmidt process, we need additional columns or information. If you provide the complete matrix, I can guide you through applying the Gram-Schmidt process to generate an orthonormal basis for its column space.