Find the QR decomposition of A=[[1,1,0],[1,0,1],[0,1,1]].
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Step-by-Step Solution
Step 1: Define the columns of A
We begin by representing the given matrix A in terms of its column vectors. This is the first step in applying the Gram-Schmidt orthogonalization process to find the QR decomposition.
Step 2: Apply Gram-Schmidt to find orthogonal vectors
We use the Gram-Schmidt process to convert the column vectors of A into a set of orthogonal vectors u1,u2,u3. The projection formula projuv=u⋅uv⋅uu is applied iteratively. For u2, we subtract the projection of a2 onto u1. For u3, we subtract the projections of a3 onto both u1 and u2.
Step 3: Normalize orthogonal vectors to find Q
To form the matrix Q, we normalize each orthogonal vector ui by dividing it by its magnitude ∥ui∥. This results in a set of orthonormal vectors qi, which become the columns of the orthogonal matrix Q.
Step 4: Calculate R using R=QTA
The matrix R can be found by multiplying the transpose of Q with the original matrix A, i.e., R=QTA. Since Q is an orthogonal matrix, QTQ=I, and thus A=QR implies QTA=QTQR=IR=R. This calculation yields an upper triangular matrix R.